  A complete Riemannian manifold without conjugate points is called
asymptotically harmonic if the mean curvature of its horospheres is a universal
constant. Examples of asymptotically harmonic manifolds include flat spaces and
rank one locally symmetric spaces of noncompact type. In this paper we show
that this list exhausts the compact asymptotically harmonic manifolds under a
variety of assumptions including nonpositive curvature or Gromov hyperbolic
fundamental group. We then present a new characterization of symmetric spaces
amongst the set of all visibility manifolds
