  We study the ergodic properties of generic continuous dynamical systems on
compact manifolds. As a main result we prove that generic homeomorphisms have
convergent Birkhoff averages under continuous observables at Lebesgue almost
every point. In spite of this, when the underlying manifold has dimension
greater than one, generic homeomorphisms have no physical measure --- a
somewhat strange result which stands in sharp contrast to current trends in
generic differentiable dynamics. Similar results hold for generic continuous
maps.
  To further explore the mysterious behaviour of $C^0$ generic dynamics, we
also study the ergodic properties of continuous maps which are conjugated to
expanding circle maps. In this context, generic maps have divergent Birkhoff
averages along orbits starting from Lebesgue almost every point.
