  We study Maker-Breaker games played on the edge set of a random graph.
Specifically, we consider the random graph process and analyze the first time
in a typical random graph process that Maker starts having a winning strategy
for his final graph to admit some property $\mP$. We focus on three natural
properties for Maker's graph, namely being $k$-vertex-connected, admitting a
perfect matching, and being Hamiltonian. We prove the following optimal hitting
time results: with high probability Maker wins the $k$-vertex connectivity game
exactly at the time the random graph process first reaches minimum degree $2k$;
with high probability Maker wins the perfect matching game exactly at the time
the random graph process first reaches minimum degree $2$; with high
probability Maker wins the Hamiltonicity game exactly at the time the random
graph process first reaches minimum degree $4$. The latter two statements
settle conjectures of Stojakovi\'{c} and Szab\'{o}.
