  This thesis has three goals related to the automorphism groups of finite
$p$-groups. The primary goal is to provide a complete proof of a theorem
showing that, in some asymptotic sense, the automorphism group of almost every
finite $p$-group is itself a $p$-group. We originally proved this theorem in a
paper with Martin; the presentation of the proof here contains omitted proof
details and revised exposition. We also give a survey of the extant results on
automorphism groups of finite $p$-groups, focusing on the order of the
automorphism groups and on known examples. Finally, we explore a connection
between automorphisms of finite $p$-groups and Markov chains. Specifically, we
define a family of Markov chains on an elementary abelian $p$-group and bound
the convergence rate of some of those chains.
