Black to move... first question, where to put the dark squared bishop?
After 4...dxc3, White often sacrifices a second pawn on b2 with 5.Bc4, but why not recapture straightaway and develop a piece with 5.Nxc3, leaving White just one pawn behind? Traditionally the main objection has been 5...Bb4, pinning the knight to the king and making it impossible for White to avoid letting it be exchanged off. However, that bishop for knight exchange does leave White with two bishops vs. bishop and knight in an open position. Black's other main options are "classical" development with 5...Bc5 and using the bishop defensively starting with 5...d6.
Meeting a premature ...Ng8-f6 with e4-e5. A key part of White's compensation in this position is that White has control over the e5-square, so an early ...Ng8-f6 can often be met effectively with e4-e5. If White has already played Bf1-c4, there are also tactics based on Bxf7+ which Black has to beware of. Thus, Black generally has to play ...d7-d6 before ...Nf6, or alternatively to develop the king's knight to e7 or h6, both of which are often sub-optimal squares.
Attacking f7 with Bc4 followed by Qb3. The idea is to force Black to make concessions in defending f7, such as by playing ...Nh6 (which develops the knight to a sub-optimal square), ...Qe7 (where the queen is vulnerable to being hit by Nc3-d5) or ...d6 and ...Qd7 (which in many cases blocks in the bishop on c8). The main drawback of this attacking plan is that White has to beware of the ...Nc6-a5 "fork", which in some cases can force the exchange of the important bishop on c4 for the knight on c6.
Attacking f7 with Bc4 followed by Ng5. White can instead play Ng5 first, which tends to be met by ...Nc6-e5 (hitting the bishop on c4), but then Black has to watch out for f2-f4, hitting the knight on e5. Black will often kick the knight on g5 with ...h6 and White will often counter with f4, opening up lines of attack for the white pieces. The main drawback of this attacking plan is that ...h6 and ...hxg5 also opens up the h-file for Black's king's rook, while meeting ...h6 with Ng5-f3 tends to slow down White's attack.
Attacking on the kingside dark squares with Bg5 and Nd5. This plan is most effective if Black has played ...Bc5 and ...d6, cutting the bishop off from defending the kingside. It is normally ineffective if Bg5 can be met by ...Be7.
Firstly, Black can pin the knight on c3 and force its exchange with 5...Bb4, which has long been regarded as the most critical line. The main drawback is that after the bishop for knight exchange, White gets two bishops vs. bishop and knight in an open position. This, together with White's open lines and development, should give White sufficient compensation for the pawn. White's most reliable reply is to hit f7 with 6.Bc4, although 6.Bg5 is an interesting alternative. Then against Black's main defence 6...d6, White's most reliable way to get compensation is to hit f7 with 7.Ng5 (instead 7.0-0 Bxc3 8.bxc3 Nf6 and 7.Qb3 Bxc3+ 8.bxc3 Qe7 9.0-0 Nf6 are less convincing). 6...Bxc3+ 7.bxc3 d6 is likewise best met by 8.Ng5.
Alternatively, Black can try using the bishop on f8 defensively with 5...d6 (5...Nf6 is inaccurate because it allows 6.e5). Sometimes dismissed because it locks the bishop on f8 inside the pawn chain, this line appears to be no better or worse than 5...Bb4. White's most reliable option is again 6.Bc4 hitting f7 (although 6.Bb5 and 6.Be2 are not clearly bad either, which aim for a more positional type of compensation using the half-open c and d-files). Then after 6...Nf6 (other moves have more drawbacks), White's most reliable way to get compensation for the pawn is to attack f7 with 7.Qb3 (here 7.Ng5 is less convincing after 7...Ne5, and 7.0-0 is well met by 7...Be7 with the plan of ...0-0 and then bringing a knight to e5 hitting the bishop on c4). Then a popular "tabiya" is reached following 7...Qd7 8.Ng5 Ne5 9.Bb5 c6 10.f4.
The other major option for Black is the Giuoco Piano-like 5...Bc5. Here again 6.Bc4 is the most reliable counter, although 6.Bg5 may also be playable. Then Black's best is 6...d6, whereupon White's immediate attacks on f7 with 7.Qb3 and 7.Ng5 don't work very well, so White should rather go for pressure on the kingside dark squares with 7.Bg5, or with 7.0-0 followed by Bg5 in most cases. The line 7.0-0 Nf6 reaches a position that can arise from the Scotch Gambit/Giuoco Piano (1.e4 e5 2.Nf3 Nc6 3.d4 exd4 4.Bc4 Bc5 5.c3 Nf6 6.0-0 dxc3 7.Nxc3 d6). This line has not been played as often but it seems that it is also critical. In particular, the line 7.Bg5 Nge7 8.Nd5 appears to be insufficient after 8...h6! (instead of 8...f6, whereupon Paul Keres's piece sacrifice 9.Bxf6 is quite dangerous) so White should rather play 8.0-0, which raises the question of whether 7.0-0 first might be more accurate as it narrows down Black's good options.
Finally I will take a look at some of Black's minor options. 5...Bd6, although rarely seen, was suggested to me by Stefan Bücker, citing Siegfried Kalkofen, and it doesn't appear to be significantly worse than the main lines. 5...Nf6 is well met by 6.e5, while 5...Be7, 5...Nge7 and 5...g6 have rather more serious drawbacks.
Books Karsten Müller & Martin Voigt, Danish Dynamite, Russel Enterprises 2003. John Emms, Play the Open Games as Black, Gambit 2000. Jude Acers, George Laven, The Italian Gambit System and a Guiding Repertoire for White - E4!, Trafford Publishing 2006.