  For a symmetric random walk in $Z^2$ which does not necessarily have bounded
jumps we study those points which are visited an unusually large number of
times. We prove the analogue of the Erd\H{o}s-Taylor conjecture and obtain the
asymptotics for the number of visits to the most visited site. We also obtain
the asymptotics for the number of points which are visited very frequently by
time $n$. Among the tools we use are Harnack inequalities and Green's function
estimates for random walks with unbounded jumps; some of these are of
independent interest.
