  In this paper, we consider the \emph{firefighter problem} on a graph
$G=(V,E)$ that is either finite or infinite. Suppose that a fire breaks out at
a given vertex $v \in V$. In each subsequent time unit, a firefighter protects
one vertex which is not yet on fire, and then the fire spreads to all
unprotected neighbors of the vertices on fire. The objective of the firefighter
is to save as many vertices as possible (if $G$ is finite) or to stop the fire
from spreading (for an infinite case).
  The surviving rate $\rho(G)$ of a finite graph $G$ is defined as the expected
percentage of vertices that can be saved when a fire breaks out at a vertex of
$G$ that is selected uniformly random. For a finite square grid $P_n \square
P_n$, we show that $5/8 + o(1) \le \rho(P_n \square P_n) \le 67243/105300 +
o(1)$ (leaving the gap smaller than 0.014) and conjecture that the surviving
rate is asymptotic to 5/8.
  We define the surviving rate for infinite graphs and prove it to be 1/4 for
the infinite square grid, even in the case of finitely many initial fires. For
the infinite hexagonal grid we provide a winning strategy if two additional
vertices can be protected at any point of the process, and we conjecture that
the firefighter has no strategy to stop the fire without additional help. We
also show how the speed of the spreading fire can be reduced by a constant
factor.
