  We extend some properties of random walks on hyperbolic groups to random
walks on convergence groups. In particular we prove that if a convergence group
$G$ acts on a compact metrizable space $M$ with the convergence property then
we can provide $G\cup M$ with a compact topology such that random walks on $G$
converge almost surely to points in $M$. Furthermore we prove that if $G$ is
finitely generated and the random walk has finite entropy and finite
logarithmic moment with respect to the word metric, then $M$, with the
corresponding hitting measure, can be seen as a model for the Poisson boundary
of $G$.
