user2001: Knight and King versus Rook and King is a theoretical draw. Yet, you might lose if you don't know the method. How do you hold a draw?

user2429: In analyzing and studying this endgame, I believe I have found a very simple way to explain the defensive technique (for this example, we will consider the defender to be Black). Once I learned this technique, I played some blitz Rook vs Knight endgames vs top chess programs and drew all of them. 
Knight's Box
I found it useful to visualize the technique in term's of a box. Consider the knight on d1 (below). The knight can be visualized in a vertical box either to its left or right as shown. Note that the box is always vertical to the base of the knight. 

In my analysis and study, I found that if Black can place its own king in the knight's box, and keep the opponent's king out of the box, then Black can hold the draw. 

Note that the knight's box only works on a non-corner box. Thus, the green boxes below work, whereas the red ones don't. 

Another exception is when the knight is pinned by the rook and the opponent king also attacks it at the same time, or when the opponent king attacks the knight and also checks the king with the rook. 

Here is a sample blitz game I played against an engine that demonstrates this technique. 
    [FEN "2R5/8/8/8/8/1K6/1n6/1k6 w - - 0 1"]
    [White "Engine"]
    [Black "Wes"] 

    1. Rb8 Nd1 2. Re8  Kc1  3. Re2 Kb1  4. Kc4  Kc1  5. Kd3 Nb2+  6. Kc3  Nd1+  
    7. Kc4  Nb2+  8. Kd4  Nd1  9. Rh2  Nb2  10. Kc3  Nd1+  11. Kb4  Nb2  
    12. Re2  Nd1  13. Ra2  Kb1  14. Kb3  Kc1  15. Rc2+  Kb1 16. Rd2  Kc1  
    17. Rh2  Kb1  18. Kc4 Kc1  19. Kd3  Nb2+ 20. Kd4  Nd1  21. Ke4  Nb2  
    22. Ke3  Nd1+  23. Kf4 Nb2  24. Ke4  Nd1  25. Rg2  Nb2  26. Kd4  Nd1  
    27. Kd3  Nb2+  28. Ke4  Nd1  29. Kd4  Nb2  30. Ke3  Nd1+  31. Kf4  Nb2  
    32. Kf3  Nd1  33. Rh2  Nb2  34. Kf4  Nd1 35. Ke5  Nb2  36. Re2 Nd1  
    37. Kd4  Nb2  38. Kd5  Nd1 39. Kc5  Nb2  40. Rh2  Nd1  41. Ra2  Nb2  
    42. Kb4  Nd1 43. Re2  Nb2  44. Kb3  Nd1 45. Ra2  Ne3  46. Kc3 Nd1+  
    47. Kd4  Nb2  48. Ke3  Nd1+  49. Kd3  Nb2+ 50. Rxb2  Kxb2 1/2-1/2

Not ALWAYS close to the King!
Although it generally helps to keep the king and knight together, it should be noted that there are some tricky positions where it is not desirable to keep the knight and the king together. This usually happens when the knight is on the b2, b7, g2 or g7 squares as shown below. 
   [FEN "2R5/8/8/8/8/1K6/1n6/1k6 w - - 0 1"]

This is an important position in Rook vs Knight endgames. Here, after White plays 1. Kc3, the move 1...Nd1, which keeps the knight somewhat close to the king, loses. Instead, 1...Na4+ draws. 
   [FEN "2R5/8/8/8/8/1K6/1n6/1k6 w - - 0 1"]

   1. Kc3 (1. Rc7 Nd1 2. Re7 Kc1) 1... Na4+ (1... Nd1+ $4 2. Kb3 Nb2 3. Kc3 Nd1+
   4. Kd2 Nb2 5. Ra8 Nc4+ 6. Kc3 Nd6 7. Ra4 Nb5+ 8. Kb3 Kc1 9. Rc4+ Kd2 10. Kb4
   Na7) 2. Kb4 (2. Kb3 Nb2 3. Rd8 Kc1!) Nb2 3. Kb3 Nd1 4. Rc7 

Here, we reach another critical position, where keeping the knight close to the king loses - 
   [FEN "2R5/8/8/8/8/1K6/8/1k1n4 w - - 0 1"]

   1. Rc7 Nf2 $3 (1...Nb2 $4 2. Rd7 Kc1 3. Rc7+ Kb1 4. Kc3 Na4+ 5. Kc4 Nb2+ 
   6. Kd4 Nd1 7. Kd3 Nb2+ 8. Kd2 Ka2 9. Kc3 Nd1+ 10. Kc2 Ne3+ 11. Kd2 Ng4 
   12. Kc3 Ne3 13. Rd7 Kb1 14. Rd3 Nf5 15. Rd1+ Ka2 16. Re1 Nd6 17. Re5 Kb1 
   18. Kb3 Kc1 19. Rc5+ Kb1 20. Rd5 Kc1 21. Rxd6) 2. Re7 Nd1 3. Rd7 (3. Re1 Kc1) 
   3... Kc1 (3... Nf2 4. Rd2 Ne4 5. Rd1#) 4. Rd8 Nb2 (4... Nf2) 5. Rc8+ Kb1 
   6. Kc3 Na4+ 7. Kb4 Nb2 8. Kb3 Nd1 1/2-1/2


user2042: The weaker side needs to keep Knight close to his King in order to achieve draw.
There are some special cases where the stronger side wins even in those situations, like when Knight is cornered or pinned in such a way that puts weaker side in zugzwang.
If the Knight is far away from the King then the result of the game depends whether or not the defending side is able to connect his pieces.
Here are some demonstrations:
Typical example:
[Title "Rook vs Knight, typical drawing plan"]
[fen "3nk3/2R5/8/4K3/8/8/8/8 w - - 0 1"] 

1.Kd6 Nf7+! ( 1...Kf8?? 2.Rc8 Ke8 3.Ra8+- ) 2.Ke6 Nd8+ 3.Kf6 Kf8 4.Rd7 Ke8 ( 4...Nc6?? 5.Rd6+- ) 5.Re7+ Kf8 6.Re1 Nb7 7.Ke6 Ke8 8.Rb1 Nd8+ 9.Kd6 Nf7+ 10.Ke6 Nd8+ 1/2-1/2 

It is useful to mention that in these position the defender should not allow vertical opposition( e6-e8 in the above example ), but should strive for the diagonal one( e8-f6 or e8/d6 ).
Even in the following position, Black is able to maintain a draw:
[Title "Drawing plan in a difficult position"]
[fen "5nk1/4R3/8/6K1/8/8/8/8 w - - 0 1"]

1.Kf6 Nh7+! 2.Kg6 Nf8+ 3.Kh6 Kh8 4.Rf7 Kg8 5.Rg7+ Kh8 6.Rg1 Nd7! ( 6...Nh7?? 7.Kg6! Kg8 8.Rg2 Nf8+ 9.Kf6+ Kh8 10.Kf7 Nh7 11.Rg8#) ( 6...Ne6 7.Kg6! Nf8 ( 7...Kg8 8.Kf6+! ) 8.Kf7 Nh7 9.Rg8# ) 7.Kg6 Kg8 8.Rd1 Nf8+ 9.Kf6 Nh7+ 10.Kg6 Nf8+ 1/2-1/2

Winning motifs for the stronger side:
We already saw one above-the decisive pinning of the knight that leads to zugzwang:
[Title "Decisive zugzwang"]
[fen "R2nk3/8/3K4/8/8/8/8/8 w - - 0 1 "]

1.Rc8 Kf7 2.Rxd8+-

These motifs happen due to bad positioning of the King:
[Title "Defenders pieces are badly positioned : Cornered King"]
[fen "6nk/5R2/8/7K/8/8/8/8 w - - 0 1 "]

1.Kg6+- ( 1.Kg5+- ) 

or badly posted Knight:
[Title "Defenders pieces are badly positioned : Knight is away from his King"]
[fen "5k2/3R4/2n2K2/8/8/8/8/8 w - - 0 1"]

1.Rd6 Ne7 2.Rd8# 1-0

Another important motif is cornering the Knight :
[Title "Cornered Knight"]
[fen "6k1/5n2/6K1/8/8/3R4/8/8 w - - 0 1 "]

1.Rd5! Nh8+ 2.Kf6 Nf7 3.Rd7 Nh6 ( 3...Nh8 4.Ra7+- ) 4.Kg6+-

Sometimes defenders units are not badly placed but just separated. In that case stronger side can win if it can push Knight away from the king since then it will be able to trap it. Here is an example:
[Title "Knight is too far from his King"]
[fen "8/8/8/2k1K3/8/3R4/4n3/8 w - - 0 1 "]

1.Re3! Ng1 2.Kf5! ( 2.Kf4? Kd4! 3.Re1 Nh3+ 4.Kg3 Ng5 5.Kf4 Nh3+= ) 2...Kd4 3.Kf4 Kc4 4.Kg3 Kd4 5.Re1+-

SUMMARY:
The defender must keep his pieces together and avoid being cornered. Maintaining diagonal opposition with kings is desired.
The stronger side has several winning motifs:

Corner the opposing King and then execute double attack with the rook which threatens mate and attacks the Knight.
Corner the Knight and create zugzwang.
Push away the Knight as far away as possible from his King and then trap him.
In some special cases, exploit bad piece coordination to create decisive zugzwang.

NOTE:
Examples and instructions are taken from the book:
Y.Averbakh - Comprehensive Chess Endings Volume 2.
Best regards.

2001: <p>Knight and King versus Rook and King is a theoretical draw. Yet, you might lose if you don't know the method. How do you hold a draw?</p>
 2429: <p>In analyzing and studying this endgame, I believe I have found a very simple way to explain the defensive technique (for this example, we will consider the defender to be Black). Once I learned this technique, I played some blitz Rook vs Knight endgames vs top chess programs and drew all of them. </p>

<p><strong>Knight's Box</strong></p>

<p>I found it useful to visualize the technique in term's of a box. Consider the knight on d1 (below). The knight can be visualized in a vertical box either to its left or right as shown. Note that the box is always vertical to the base of the knight. </p>

<p><img src="https://i.stack.imgur.com/zaGCJ.jpg" alt="enter image description here"></p>

<p>In my analysis and study, I found that if Black can place its own king in the knight's box, and keep the opponent's king out of the box, then Black can hold the draw. </p>

<p><img src="https://i.stack.imgur.com/1TaXs.jpg" alt="enter image description here"></p>

<p>Note that the knight's box only works on a <strong>non-corner</strong> box. Thus, the green boxes below work, whereas the red ones don't. </p>

<p><img src="https://i.stack.imgur.com/wv82o.jpg" alt="enter image description here"></p>

<p>Another exception is when the knight is pinned by the rook and the opponent king also attacks it at the same time, or when the opponent king attacks the knight and also checks the king with the rook. </p>

<p><img src="https://i.stack.imgur.com/9O2NY.jpg" alt="enter image description here"></p>

<p>Here is a sample blitz game I played against an engine that demonstrates this technique. </p>

<pre><code>    [FEN "2R5/8/8/8/8/1K6/1n6/1k6 w - - 0 1"]
    [White "Engine"]
    [Black "Wes"] 

    1. Rb8 Nd1 2. Re8  Kc1  3. Re2 Kb1  4. Kc4  Kc1  5. Kd3 Nb2+  6. Kc3  Nd1+  
    7. Kc4  Nb2+  8. Kd4  Nd1  9. Rh2  Nb2  10. Kc3  Nd1+  11. Kb4  Nb2  
    12. Re2  Nd1  13. Ra2  Kb1  14. Kb3  Kc1  15. Rc2+  Kb1 16. Rd2  Kc1  
    17. Rh2  Kb1  18. Kc4 Kc1  19. Kd3  Nb2+ 20. Kd4  Nd1  21. Ke4  Nb2  
    22. Ke3  Nd1+  23. Kf4 Nb2  24. Ke4  Nd1  25. Rg2  Nb2  26. Kd4  Nd1  
    27. Kd3  Nb2+  28. Ke4  Nd1  29. Kd4  Nb2  30. Ke3  Nd1+  31. Kf4  Nb2  
    32. Kf3  Nd1  33. Rh2  Nb2  34. Kf4  Nd1 35. Ke5  Nb2  36. Re2 Nd1  
    37. Kd4  Nb2  38. Kd5  Nd1 39. Kc5  Nb2  40. Rh2  Nd1  41. Ra2  Nb2  
    42. Kb4  Nd1 43. Re2  Nb2  44. Kb3  Nd1 45. Ra2  Ne3  46. Kc3 Nd1+  
    47. Kd4  Nb2  48. Ke3  Nd1+  49. Kd3  Nb2+ 50. Rxb2  Kxb2 1/2-1/2
</code></pre>

<p><strong>Not ALWAYS close to the King!</strong></p>

<p>Although it generally helps to keep the king and knight together, it should be noted that there are some tricky positions where it is <strong>not desirable</strong> to keep the knight and the king together. This usually happens when the knight is on the <strong>b2</strong>, <strong>b7</strong>, <strong>g2</strong> or <strong>g7</strong> squares as shown below. </p>

<pre><code>   [FEN "2R5/8/8/8/8/1K6/1n6/1k6 w - - 0 1"]
</code></pre>

<p>This is an important position in Rook vs Knight endgames. Here, after White plays <strong>1. Kc3</strong>, the move <strong>1...Nd1</strong>, which keeps the knight somewhat close to the king, loses. Instead, <strong>1...Na4+</strong> draws. </p>

<pre><code>   [FEN "2R5/8/8/8/8/1K6/1n6/1k6 w - - 0 1"]

   1. Kc3 (1. Rc7 Nd1 2. Re7 Kc1) 1... Na4+ (1... Nd1+ $4 2. Kb3 Nb2 3. Kc3 Nd1+
   4. Kd2 Nb2 5. Ra8 Nc4+ 6. Kc3 Nd6 7. Ra4 Nb5+ 8. Kb3 Kc1 9. Rc4+ Kd2 10. Kb4
   Na7) 2. Kb4 (2. Kb3 Nb2 3. Rd8 Kc1!) Nb2 3. Kb3 Nd1 4. Rc7 
</code></pre>

<p>Here, we reach another critical position, where keeping the knight close to the king loses - </p>

<pre><code>   [FEN "2R5/8/8/8/8/1K6/8/1k1n4 w - - 0 1"]

   1. Rc7 Nf2 $3 (1...Nb2 $4 2. Rd7 Kc1 3. Rc7+ Kb1 4. Kc3 Na4+ 5. Kc4 Nb2+ 
   6. Kd4 Nd1 7. Kd3 Nb2+ 8. Kd2 Ka2 9. Kc3 Nd1+ 10. Kc2 Ne3+ 11. Kd2 Ng4 
   12. Kc3 Ne3 13. Rd7 Kb1 14. Rd3 Nf5 15. Rd1+ Ka2 16. Re1 Nd6 17. Re5 Kb1 
   18. Kb3 Kc1 19. Rc5+ Kb1 20. Rd5 Kc1 21. Rxd6) 2. Re7 Nd1 3. Rd7 (3. Re1 Kc1) 
   3... Kc1 (3... Nf2 4. Rd2 Ne4 5. Rd1#) 4. Rd8 Nb2 (4... Nf2) 5. Rc8+ Kb1 
   6. Kc3 Na4+ 7. Kb4 Nb2 8. Kb3 Nd1 1/2-1/2
</code></pre>
 2042: <p>The weaker side needs to keep Knight <strong>close to his King</strong> in order to achieve draw.</p>

<p>There are some special cases where the stronger side wins even in those situations, like when Knight is cornered or pinned in such a way that puts weaker side in <em>zugzwang</em>.</p>

<p>If the Knight is far away from the King then the result of the game depends whether or not the defending side is able to connect his pieces.</p>

<p>Here are some demonstrations:</p>

<h2>Typical example:</h2>

<pre><code>[Title "Rook vs Knight, typical drawing plan"]
[fen "3nk3/2R5/8/4K3/8/8/8/8 w - - 0 1"] 

1.Kd6 Nf7+! ( 1...Kf8?? 2.Rc8 Ke8 3.Ra8+- ) 2.Ke6 Nd8+ 3.Kf6 Kf8 4.Rd7 Ke8 ( 4...Nc6?? 5.Rd6+- ) 5.Re7+ Kf8 6.Re1 Nb7 7.Ke6 Ke8 8.Rb1 Nd8+ 9.Kd6 Nf7+ 10.Ke6 Nd8+ 1/2-1/2 
</code></pre>

<p>It is useful to mention that in these position the defender <strong>should not allow vertical opposition</strong>( <code>e6-e8</code> in the above example ), but <strong>should strive for the diagonal one</strong>( <code>e8-f6</code> or <code>e8/d6</code> ).</p>

<p>Even in the following position, Black is able to maintain a draw:</p>

<pre><code>[Title "Drawing plan in a difficult position"]
[fen "5nk1/4R3/8/6K1/8/8/8/8 w - - 0 1"]

1.Kf6 Nh7+! 2.Kg6 Nf8+ 3.Kh6 Kh8 4.Rf7 Kg8 5.Rg7+ Kh8 6.Rg1 Nd7! ( 6...Nh7?? 7.Kg6! Kg8 8.Rg2 Nf8+ 9.Kf6+ Kh8 10.Kf7 Nh7 11.Rg8#) ( 6...Ne6 7.Kg6! Nf8 ( 7...Kg8 8.Kf6+! ) 8.Kf7 Nh7 9.Rg8# ) 7.Kg6 Kg8 8.Rd1 Nf8+ 9.Kf6 Nh7+ 10.Kg6 Nf8+ 1/2-1/2
</code></pre>

<h2>Winning motifs for the stronger side:</h2>

<p>We already saw one above-the decisive pinning of the knight that leads to <em>zugzwang</em>:</p>

<pre><code>[Title "Decisive zugzwang"]
[fen "R2nk3/8/3K4/8/8/8/8/8 w - - 0 1 "]

1.Rc8 Kf7 2.Rxd8+-
</code></pre>

<p>These motifs happen <strong>due to bad positioning</strong> of the King:</p>

<pre><code>[Title "Defenders pieces are badly positioned : Cornered King"]
[fen "6nk/5R2/8/7K/8/8/8/8 w - - 0 1 "]

1.Kg6+- ( 1.Kg5+- ) 
</code></pre>

<p>or badly posted Knight:</p>

<pre><code>[Title "Defenders pieces are badly positioned : Knight is away from his King"]
[fen "5k2/3R4/2n2K2/8/8/8/8/8 w - - 0 1"]

1.Rd6 Ne7 2.Rd8# 1-0
</code></pre>

<p>Another important motif is <strong>cornering the Knight</strong> :</p>

<pre><code>[Title "Cornered Knight"]
[fen "6k1/5n2/6K1/8/8/3R4/8/8 w - - 0 1 "]

1.Rd5! Nh8+ 2.Kf6 Nf7 3.Rd7 Nh6 ( 3...Nh8 4.Ra7+- ) 4.Kg6+-
</code></pre>

<p>Sometimes defenders units are not badly placed but just separated. In that case stronger side can win if it can push Knight away from the king since then it will be able to trap it. Here is an example:</p>

<pre><code>[Title "Knight is too far from his King"]
[fen "8/8/8/2k1K3/8/3R4/4n3/8 w - - 0 1 "]

1.Re3! Ng1 2.Kf5! ( 2.Kf4? Kd4! 3.Re1 Nh3+ 4.Kg3 Ng5 5.Kf4 Nh3+= ) 2...Kd4 3.Kf4 Kc4 4.Kg3 Kd4 5.Re1+-
</code></pre>

<h2>SUMMARY:</h2>

<p>The defender must keep his pieces together and avoid being cornered. Maintaining <strong>diagonal</strong> opposition with kings is desired.</p>

<p>The stronger side has several winning motifs:</p>

<ol>
<li><p>Corner the opposing King and then execute double attack with the rook which threatens mate and attacks the Knight.</p></li>
<li><p>Corner the Knight and create <em>zugzwang</em>.</p></li>
<li><p>Push away the Knight as far away as possible from his King and then trap him.</p></li>
<li><p>In some special cases, exploit bad piece coordination to create decisive <em>zugzwang</em>.</p></li>
</ol>

<p><em>NOTE:</em></p>

<p>Examples and instructions are taken from the book:</p>

<p><em>Y.Averbakh - Comprehensive Chess Endings Volume 2</em>.</p>

<p>Best regards.</p>
