  In a Maker-Breaker game on a graph $G$, Breaker and Maker alternately claim
edges of $G$. Maker wins if, after all edges have been claimed, the graph
induced by his edges has some desired property. We consider four Maker-Breaker
games played on random geometric graphs. For each of our four games we show
that if we add edges between $n$ points chosen uniformly at random in the unit
square by order of increasing edge-length then, with probability tending to one
as $n\to\infty$, the graph becomes Maker-win the very moment it satisfies a
simple necessary condition. In particular, with high probability, Maker wins
the connectivity game as soon as the minimum degree is at least two; Maker wins
the Hamilton cycle game as soon as the minimum degree is at least four; Maker
wins the perfect matching game as soon as the minimum degree is at least two
and every edge has at least three neighbouring vertices; and Maker wins the
$H$-game as soon as there is a subgraph from a finite list of "minimal graphs".
These results also allow us to give precise expressions for the limiting
probability that $G(n,r)$ is Maker-win in each case, where $G(n,r)$ is the
graph on $n$ points chosen uniformly at random on the unit square with an edge
between two points if and only if their distance is at most $r$.
