  Random walks on a group $G$ model many natural phenomena. A random walk is
defined by a probability measure $p$ on $G$. We are interested in asymptotic
properties of the random walks and in particular in the linear drift and the
asymptotic entropy. If the geometry of the group is rich, then these numbers
are both positive and the way of dependence on $p$ is itself a property of $G$.
In this note, we review recent results about the regularity of the drift and
the entropy for free groups, free products and hyperbolic groups.
