  The Lichnerowicz conjecture asserts that all harmonic manifolds are either
flat or locally symmetric spaces of rank 1. This conjecture has been proved by
Z.I. Szabo for harmonic manifolds with compact universal cover. E. Damek and F.
Ricci provided examples showing that in the noncompact case the conjecture is
wrong. However, such manifolds do not admit a compact quotient. The
classification of all noncompact harmonic spaces is still a very difficult open
problem.
  In this paper we provide a survey on recent results on noncompact simply
connected harmonic manifolds, and we also prove many new results, both for
general noncompact harmonic manifolds and for noncompact harmonic manifolds
with purely exponential volume growth.
