user15223: For this question I'm interested only in gambits that:

Are real gambits. That is, the side that's risking the material actually ends up down material + cannot count on winning the material back. This, e.g., excludes the Queen's Gambit because 1. d4 d5 2. c4 dxc4 3. Qa4+ wins the material back without problems (plus most Queen's Gambit lines don't end with White down material). Also, if the gambit cannot be safely accepted, it's excluded.
Don't rely on opponent making a mistake. So the Fried Liver Attack 1. e4 e5 2. Nf3 Nc6 3. Bc4 Nf6 4. Ng5 d5 5. exd5 Nxd5?! 6. Nxf7 doesn't count since 5...Nxd5 is a well-known bad move.

I notice that Stockfish doesn't seem to like gambits. Analyzing using the engine at www.chess.com/analysis, for example:

King's Gambit: 1. e4 e5 2. f4 yields a starting eval of -1.10 (at depth 19).
Evan's Gambit: 1. e4 e5 2. Nf3 Nc6 3. Bc4 Bc5 4. b4 yields a starting eval of -0.60 (d =19).
Danish Gambit: 1. e4 e5 2. d4 exd4 3. c3 yields a starting eval of -0.48.
Sicilian Wing Gambit: 1. e4 c5 2. b4 yields a starting eval of -0.44.
Smith Morra Gambit: 1. e4 c5 2. d4 cxd4 3. c3 yields a starting eval of -0.19
Blackmar-Diemer Gambit: 1. d4 d5 2. Nc3 Nf6 3. e4 dxe4 4. f3 yields a starting eval of -0.49.

(I'd consider Black gambits as well but those are harder to evaluate since the starting position favors White, and therefore it's not surprising if White has an advantage after Black plays the gambit)
Are there any gambits in which the side that is down material for tempo is advantaged? 

user23908: the fried liver is one of the 2 only real gambit stockfish likes, or the queens gambit. it appears that the benko gambit is one gambit that have big pressure against whites Q side

user15478: A variation of the Caro Kann, Advance: Tal Variation called Caveman Variation is a "gambit" where White can gambit mere pawns, or whole pieces, but probably will mate. If Black should go for the rook, the imbalance most often is 2 rooks + 2 pawns for the queen in a structure that favors the queen.
[FEN ""]
1. e4 c6 2. d4 d5 3. e5 Bf5 4. h4 h5
5. Bg5 Qb6 6. Bd3 Bxd3 7. Qxd3 Qxb2 8. e6

SF11 at depth ~40 says +0.8, but if you look at human games, White win percentage is huge. There is 6...Qxd4, but play is very artificial, most players who decline do so with 7...e6 or trying to trade queens immediately.

user18809: You can play the Queen's gambit if you go for 3.Nf3, actually offering your opponent a chance to stick to the pawn later on, so even if it's not a "real gambit" at move 2, you can turn it into one later on. The position often gives white a small advantage.
Another interesting possibility is the From's Gambit (1.f4 e5 fxe5 d6 exd6 Bxd6) which is often said to at least equalize (not bad being Black)
Anyway, I wouldn't trust the engine's evaluation in the opening (if we did, we'd be playing 1.Nc3 all day long!) If the Danish Gambit really leads to -0.48 while beign two pawns down, it probably means White is actually better unless there is some good obvious continuation for Black!. And this is assuming the engine's opinion is "not that wrong". For example, the Evans Gambit is not played because Black can equalize, not because it can get a -0.6 position.
By the way, what about the Scotch Gambit?

user15223: Games 61/62 of the 17th TCEC season featured such a gambit:
[FEN ""]
1. d4 Nf6 2. Bg5 c5 3. d5 Qb6 4. Nc3

White gambits the b2-pawn. After 4...Qxb2 5. Bd2, Stockfish evaluated the position as +0.61 for White. In the reverse game, Leela gave White a +0.49 advantage after the same moves.

user20527: While it's a gambit by Black, what about Tal's gambit?  Black is scoring 55% in 228 grandmaster and elite correspondence games after 1 e4 c5; 2 f4 d5; 3 exd5 Nf6.  I'd call Black doing better than 50% in that many top games an advantage.  (And 55% is better than Black's score in any of the main non-gambit responses to 2 f4.)

15223: <p>For this question I'm interested only in gambits that:</p>

<ol>
<li>Are real gambits. That is, the side that's risking the material actually ends up down material + cannot count on winning the material back. This, e.g., excludes the Queen's Gambit because 1. d4 d5 2. c4 dxc4 3. Qa4+ wins the material back without problems (plus most Queen's Gambit lines don't end with White down material). Also, if the gambit cannot be safely accepted, it's excluded.</li>
<li>Don't rely on opponent making a mistake. So the Fried Liver Attack 1. e4 e5 2. Nf3 Nc6 3. Bc4 Nf6 4. Ng5 d5 5. exd5 Nxd5?! 6. Nxf7 doesn't count since 5...Nxd5 is a well-known bad move.</li>
</ol>

<p>I notice that Stockfish doesn't seem to like gambits. Analyzing using the engine at www.chess.com/analysis, for example:</p>

<ul>
<li>King's Gambit: 1. e4 e5 2. f4 yields a starting eval of -1.10 (at depth 19).</li>
<li>Evan's Gambit: 1. e4 e5 2. Nf3 Nc6 3. Bc4 Bc5 4. b4 yields a starting eval of -0.60 (d =19).</li>
<li>Danish Gambit: 1. e4 e5 2. d4 exd4 3. c3 yields a starting eval of -0.48.</li>
<li>Sicilian Wing Gambit: 1. e4 c5 2. b4 yields a starting eval of -0.44.</li>
<li>Smith Morra Gambit: 1. e4 c5 2. d4 cxd4 3. c3 yields a starting eval of -0.19</li>
<li>Blackmar-Diemer Gambit: 1. d4 d5 2. Nc3 Nf6 3. e4 dxe4 4. f3 yields a starting eval of -0.49.</li>
</ul>

<p>(I'd consider Black gambits as well but those are harder to evaluate since the starting position favors White, and therefore it's not surprising if White has an advantage after Black plays the gambit)</p>

<p>Are there any gambits in which the side that is down material for tempo is advantaged? </p>
 23908: <p>the fried liver is one of the 2 only real gambit stockfish likes, or the queens gambit. it appears that the benko gambit is one gambit that have big pressure against whites Q side</p>
 15478: <p>A variation of the Caro Kann, Advance: Tal Variation called Caveman Variation is a &quot;gambit&quot; where White can gambit mere pawns, or whole pieces, but probably will mate. If Black should go for the rook, the imbalance most often is 2 rooks + 2 pawns for the queen in a structure that favors the queen.</p>
<pre><code>[FEN &quot;&quot;]
1. e4 c6 2. d4 d5 3. e5 Bf5 4. h4 h5
5. Bg5 Qb6 6. Bd3 Bxd3 7. Qxd3 Qxb2 8. e6
</code></pre>
<p>SF11 at depth ~40 says +0.8, but if you look at human games, White win percentage is huge. There is 6...Qxd4, but play is very artificial, most players who decline do so with 7...e6 or trying to trade queens immediately.</p>
 18809: <p>You can play the Queen's gambit if you go for 3.Nf3, actually offering your opponent a chance to stick to the pawn later on, so even if it's not a "real gambit" at move 2, you can turn it into one later on. The position often gives white a small advantage.</p>

<p>Another interesting possibility is the From's Gambit (1.f4 e5 fxe5 d6 exd6 Bxd6) which is often said to at least equalize (not bad being Black)</p>

<p>Anyway, I wouldn't trust the engine's evaluation in the opening (if we did, we'd be playing 1.Nc3 all day long!) If the Danish Gambit really leads to -0.48 while beign two pawns down, it probably means White is actually better unless there is some good obvious continuation for Black!. And this is assuming the engine's opinion is "not that wrong". For example, the Evans Gambit is not played because Black can equalize, not because it can get a -0.6 position.</p>

<p>By the way, what about the Scotch Gambit?</p>
 15223: <p>Games 61/62 of the 17th TCEC season featured such a gambit:</p>

<pre><code>[FEN ""]
1. d4 Nf6 2. Bg5 c5 3. d5 Qb6 4. Nc3
</code></pre>

<p>White gambits the b2-pawn. After 4...Qxb2 5. Bd2, Stockfish evaluated the position as <a href="https://www.tcec-chess.com/archive.html?season=17&amp;div=sf&amp;game=61" rel="noreferrer">+0.61 for White</a>. In <a href="https://www.tcec-chess.com/archive.html?season=17&amp;div=sf&amp;game=62" rel="noreferrer">the reverse game</a>, Leela gave White a +0.49 advantage after the same moves.</p>
 20527: <p>While it's a gambit by Black, what about Tal's gambit?  Black is scoring 55% in 228 grandmaster and elite correspondence games after 1 e4 c5; 2 f4 d5; 3 exd5 Nf6.  I'd call Black doing better than 50% in that many top games an advantage.  (And 55% is better than Black's score in any of the main non-gambit responses to 2 f4.)</p>
