This includes coverage of the three most important ways of declining the gambit: 1.e4 e5 2.f4 Bc5, 2...d5, and 2...Nc6. Until a few months ago I had not really examined 2...Bc5 and so didn't really know what I was doing when I had it a few times from the White side.
Some comments on individual lines
The line 2...d5 3.exd5 e4 appears to be slightly better than I had previously thought. Some of Boris Alterman's ideas, e.g. in the lines 4.d3 Nf6 5.dxe4 Nxe4 6.Be3 Bd6!?, and 6.Nf3 c6!?, appear to offer Black reasonable practical chances, although I agree with John Shaw that White should be able to get an advantage with accurate play in all lines. White also appears to be slightly better against 3...c6, so Black's objectively best follow-up to 2...d5 is 3.exd5 exf4, which usually leads to a Modern Defence (2...exf4 3.Nf3 d5) but without allowing 3.Bc4 d5 4.Bxd5. My latest examinations of that line suggest that Black might be able to equalise with accurate play but I find the positions more appealing for White than after 3.Nf3 d5.
The most critical test of 3.Bc4 remains 3...Nc6. The most critical line runs 4.d4 Nf6 5.Nc3 Bb4 6.Nge2 f3 7.gxf3 d5 8.exd5 Nxd5 9.0-0 Nxc3 10.bxc3 Bd6.
White has the inferior pawn structure and an exposed king, but has plenty of open lines for the bishops and rooks. As White I would be tempted to play Kh1 and Rg1 and make use of the half-open g-file, but it is questionable whether White can make much of this. Probably objectively best is Stefan Bucker's suggestion 11.Qd2 intending to encourage a queen exchange with Qg5, and thus making White's exposed king less of a problem, My analysis then runs 11...0-0 12.Qg5 Qxg5+ 13.Bxg5 Bf5 14.Bb3 Na5 15.Ng3 Nxb3 16.axb3. The queenless middlegame gives approximately equal chances and the rival pawn majorities ensure that there is plenty of play left. In view of this, I disagree with John Shaw's claim that 3...Nc6 is a refutation of 3.Bc4, but I think that the resulting positions are generally easier for Black to play than for White.
The assessment of this line lies somewhere in the grey area between "=" and "+=" and White tends to get most of the attacking chances, although it is probably Black's best way of getting a fairly closed position with level material against the King's Gambit. If Black's aim is rather just to reach dynamically equal positions with level material then the Modern Defence is a better bet, and if Black has a problem with 3.Bc4 d5 4.Bxd5 then there is 2...d5 3.exd5 exf4.
There is one line recommended by Mihail Marin which has left me wondering:
3.Nf3 d6 4.Nc3 Nf6 5.Bc4 Nc6 6.d3 Bg4 7.Na4 0-0 8.Nxc5 dxc5 9.0-0 Qd6 10.f5 Nd4 11.a4 a6 12.c3 b5 13.cxd4 Bxf3
Fedorov-Marin, Eforie Nord 2000 continued with 14.Qxf3 bxc4 here and the game was soon agreed drawn, but why not 14.Bxf7+ followed by 15.Rxf3? I have to admit that Fritz spotted this before I did. It looks quite promising for White.
The sideline 4...Nc6 intending 5.Bc4 Bg4 looks like it may improve slightly over the main line for Black, but White has the option of changing plans with 5.Bb5, whereupon Black's best seems to be to sacrifice a pawn for compensation with 5...Nge7 6.Na4 Bg4 7.fxe5 0-0 8.exd6 Bxd6.
White can avoid all of this with 4.c3, which works well unless Black finds 4...Bb6! The line appears to offer equal chances in a double-edged position after 5.d4 exd4 6.cxd4 Bg4 7.Be3, with the idea of h2-h3 and meeting ...Bg4xf3 with the daring g2xf3, hoping to use the impressive pawn centre to compensate for long-term issues with king safety. 4.c3 thus remains playable but less likely to provide a theoretical edge for White than 4.Nc3.