  The aim of the present paper is to study conditions under which all the
non-wandering points are periodic points, for a discrete dynamical system of
two variables defined on a compact manifold. We include a survey of known
results in all dimensions, and study the remaining open question in dimension
two. We present two results, one positive and one negative. The negative
result: we construct a Kupka--Smale diffeomorphism in $\mathbb{R}^2$ (which can
be extended to a diffeomorphism of the sphere) with a closed set of periodic
points that differs from the set of nonwandering points. The positive result:
we present a condition on the widely studied H\'{e}non family which guarantees
that all nonwandering points are periodic. Finally, we close by describing what
future work may be needed to resolve our broad goals.
