  In this note, we prove that a random extension of either the free group $F_N$
of rank $N\ge3$ or of the fundamental group of a closed, orientable surface
$S_g$ of genus $g\ge2$ is a hyperbolic group. Here, a random extension is one
corresponding to a subgroup of either Out$(F_N)$ or Mod$(S_g)$ generated by $k$
independent random walks. Our main theorem has several applications, including
that a random subgroup of a weakly hyperbolic group is free and undistorted.
