The Hogeye Gambit is actually a move-order rather than an independent opening. It is a forcing way to get into an Italian Game - that is a Giuoco Piano or Two Knights Defense - without allowing specialty defenses to 1 e4 e5 2 Nf3. It not only avoids counter-attacks like the Latvian countergambit and Elephant Gambit, but also black's preferred lines in the Philador and Petroff defenses.

There are three parts to the Hogeye Gambit.

Early deviations and the Scotch Gambit The Two Knights Defense - Modern Variation The Giuoco Piano - Steinitz-Sveshnikov Variation

There other ways of playing against the Two Knights (like Morphy's 5 0-0) and the Giuoco (6 Bd2 with an IQP game, or 6 Nc3 the Moller Attack), so these lines can be matched to your own repertoire.