We initiate the study of the algorithmic complexity of Maker-Breaker games played on edge sets of graphs for general graphs. We mainly consider three of the big four such games: the connectivity game, perfect matching game, and H-game. Maker wins if she claims the edges of a spanning tree in the first, a perfect matching in the second, and a copy of a fixed graph H in the third. We prove that deciding who wins the perfect matching game and the H-game is PSPACE-complete, even for the latter in graphs of small diameter if H is a tree. Seeking to find the smallest graph H such that the H-game is PSPACE-complete, we also prove that there exists such an H of order 51 and size 57. On the positive side, we show that the connectivity game and arboricity-k game are polynomial-time solvable. We then give several positive results for the H-game, first giving a structural characterization for Breaker to win the P 4-game, which gives a linear-time algorithm for the P 4-game. We provide a structural characterization for Maker to win the K 1,ℓ-game in trees, which implies a linear-time algorithm for the K 1,ℓ-game in trees. Lastly, we prove that the K 1,ℓ-game in any graph, and the H-game in trees are both FPT parameterized by the length of the game. We leave the complexity of the last of the big four games, the Hamiltonicity game, as an open question.