user269: Suppose we have a rook and the opponent has a knight next to his king.
Is it possible to win this game theoretically?  
E.g., in the following position (white to move), is it possible for white to win?
[FEN "8/8/8/3n4/3k4/8/K7/R7 w - - 0 1"]


user17: If the knight is close to the king, it is hard to win with a rook. That's because the knight will fend off the enemy pieces that try to approach the king. Most winning positions feature the knight and king far apart so that the opposing king and rook can pick them off one by one (checkmate the king or trap the knight).
An important exception to the above is if the knight is either on a corner square, or is a "knights jump" from the corner square. A knight has only two moves from a corner square, meaning it may not have enough maneuvering room. If the knight can find a safe haven on the open board it can defend its king.
A similar thing is true with a bishop and king against a rook and king.

user4454: If it's a normal game, a rook against a knight will always be a draw.
In very rare situation both can win:
A knight wins here in this position
 [Title "White to move"]   
 [FEN "8/8/8/8/8/8/r2N4/k1K5 w - - 0 1"]

A rook wins here in this position
 [Title "White to move"]   
 [FEN "8/2R5/8/8/7k/3K4/8/4n3 w - - 0 1"]


user16394: I believe it's a draw given that the king and knight stick to each other. Someone has already mentioned the "Nalimov Tablebases". Give it a try and explore the variations.
However, I just lost a game with the knight (the reason why I was searching for this question XD). I was really low on time and voluntarily moved my king to a square where my knight was pinned. After playing around with the Nalimov Tablebases, here are some tips for defending the draw:

Do not move your knight away from your king.
Be care of pins and skewers.
Try to stay at the center. (Someone pointed out that you can defend a draw with your knight on b1 and king on c1. But staying near the center allows more room for error).


user12: In the specific position that you mention, the answer is a resounding no.  The king and the knight will defend each other, and white will not be able to force mate.
However, the knight is a clumsy piece.  If the knight is not positioned perfectly, then the rook will frequently be able to drive the knight to a bad square and deprive it of moves.  Eventually, the knight may be captured, or the king might be checkmated.
As an example of a position where the rook can win, take the following: 
[FEN "8/1n6/8/8/2R2K2/8/5k2/8 w - - 0 1"]
[SetUp "1"]
[PlyCount "17"]

1. Rc2+ Ke1 2. Ke3 Kd1 (2... Kf1 3. Rf2+ Kg1 4. Rd2 Nc5 5. Kf3 Ne6 6. Kg3 Kf1
7. Rd5 Nc7 8. Re5 Na6 9. Kf3 Kg1 10. Rg5+ Kf1 11. Ra5) 3. Rb2 Nc5 4. Rb6 $22
Na4 5. Rb4 Nc3 (5... Nc5 6. Rd4+ Kc2 7. Rc4+) 6. Kd3 Na2 7. Rb1+ Nc1+ 8. Kc3
Ke2 9. Rxc1 *

Here, white can win starting with 1.Rc2+.  The plan is to drive the king to the first rank and then attack the knight while improving the position of the rook.  

In your question, you mention specific cases where the king and the knight are close together.  In these cases, it is very important to know two key positions:
First, in this position, black can hold the draw:

No matter what white tries, the white king cannot approach the black king due to the knight.  Black will be able to squeeze out with ...Kb2.  Black's plan is to mark time by playing ...Na3 and ...Nb1.  White cannot make useful progress.
This position, almost the same, has one major difference - black can no longer mark time with the knight:

If white plays a waiting move, for example 1.Re2, then black must lose immediately.  Any move will lose the knight or allow mate in one.

If you're interested in a specific position, you can check out the Nalimov Tablebases online.  They have completely solved every position with 6 or fewer pieces on the board.

269: <p>Suppose we have a rook and the opponent has a knight next to his king.</p>

<p>Is it possible to win this game theoretically?  </p>

<p>E.g., in the following position (white to move), is it possible for white to win?</p>

<pre><code>[FEN "8/8/8/3n4/3k4/8/K7/R7 w - - 0 1"]
</code></pre>
 17: <p>If the knight is close to the king, it is hard to win with a rook. That's because the knight will fend off the enemy pieces that try to approach the king. Most winning positions feature the knight and king far apart so that the opposing king and rook can pick them off one by one (checkmate the king or trap the knight).</p>

<p>An important exception to the above is if the knight is either on a corner square, or is a "knights jump" from the corner square. A knight has only two moves from a corner square, meaning it may not have enough maneuvering room. If the knight can find a safe haven on the open board it can defend its king.</p>

<p>A similar thing is true with a bishop and king against a rook and king.</p>
 4454: <p>If it's a normal game, a rook against a knight will always be a draw.</p>

<p>In very rare situation both can win:</p>

<p>A knight wins here in this position</p>

<pre><code> [Title "White to move"]   
 [FEN "8/8/8/8/8/8/r2N4/k1K5 w - - 0 1"]
</code></pre>

<p>A rook wins here in this position</p>

<pre><code> [Title "White to move"]   
 [FEN "8/2R5/8/8/7k/3K4/8/4n3 w - - 0 1"]
</code></pre>
 16394: <p>I believe it's a draw given that the king and knight stick to each other. Someone has already mentioned the "Nalimov Tablebases". Give it a try and explore the variations.
However, I just lost a game with the knight (the reason why I was searching for this question XD). I was really low on time and voluntarily moved my king to a square where my knight was pinned. After playing around with the Nalimov Tablebases, here are some tips for defending the draw:</p>

<ol>
<li>Do not move your knight away from your king.</li>
<li>Be care of pins and skewers.</li>
<li>Try to stay at the center. (Someone pointed out that you can defend a draw with your knight on b1 and king on c1. But staying near the center allows more room for error).</li>
</ol>
 12: <p>In the specific position that you mention, the answer is a resounding <strong>no</strong>.  The king and the knight will defend each other, and white will not be able to force mate.</p>

<p>However, the knight is a clumsy piece.  If the knight is not positioned perfectly, then the rook will frequently be able to drive the knight to a bad square and deprive it of moves.  Eventually, the knight may be captured, or the king might be checkmated.</p>

<p>As an example of a position where the rook can win, take the following: </p>

<pre><code>[FEN "8/1n6/8/8/2R2K2/8/5k2/8 w - - 0 1"]
[SetUp "1"]
[PlyCount "17"]

1. Rc2+ Ke1 2. Ke3 Kd1 (2... Kf1 3. Rf2+ Kg1 4. Rd2 Nc5 5. Kf3 Ne6 6. Kg3 Kf1
7. Rd5 Nc7 8. Re5 Na6 9. Kf3 Kg1 10. Rg5+ Kf1 11. Ra5) 3. Rb2 Nc5 4. Rb6 $22
Na4 5. Rb4 Nc3 (5... Nc5 6. Rd4+ Kc2 7. Rc4+) 6. Kd3 Na2 7. Rb1+ Nc1+ 8. Kc3
Ke2 9. Rxc1 *
</code></pre>

<p>Here, white can win starting with <code>1.Rc2+</code>.  The plan is to drive the king to the first rank and then attack the knight while improving the position of the rook.  </p>

<hr />

<p>In your question, you mention specific cases where the king and the knight are close together.  In these cases, it is very important to know two key positions:</p>

<p>First, in this position, black can hold the draw:<br>
<img src="https://i.stack.imgur.com/5wZlu.png" alt="Knight vs Rook Endgame where black holds the draw"><br>
No matter what white tries, the white king cannot approach the black king due to the knight.  Black will be able to squeeze out with <code>...Kb2</code>.  Black's plan is to mark time by playing <code>...Na3</code> and <code>...Nb1</code>.  White cannot make useful progress.</p>

<p>This position, almost the same, has one major difference - black can no longer mark time with the knight:<br>
<img src="https://i.stack.imgur.com/LwSxo.png" alt="Knight vs Rook Endgame where white wins"><br>
If white plays a waiting move, for example <code>1.Re2</code>, then black must lose immediately.  Any move will lose the knight or allow mate in one.</p>

<hr />

<p>If you're interested in a specific position, you can check out the Nalimov Tablebases <a href="http://www.k4it.de/index.php?topic=egtb&amp;lang=en" rel="noreferrer">online</a>.  They have completely solved every position with 6 or fewer pieces on the board.</p>
