  We study the Maker-Breaker $H$-game played on the edge set of the random
graph $G_{n,p}$. In this game two players, Maker and Breaker, alternately claim
unclaimed edges of $G_{n,p}$, until all the edges are claimed. Maker wins if he
claims all the edges of a copy of a fixed graph $H$; Breaker wins otherwise. In
this paper we show that, with the exception of trees and triangles, the
threshold for an $H$-game is given by the threshold of the corresponding Ramsey
property of $G_{n,p}$ with respect to the graph $H$.
