  In this paper we prove two results. First we show that dynamical systems with
a $\phi$-mixing measure have in the limit Poisson distributed return times
almost everywhere. We use the Chen-Stein method to also obtain rates of
convergence. Our theorem improves on previous results by allowing for infinite
partitions and dropping the requirement that the invariant measure have finite
entropy with respect to the given partition. As has been shown elsewhere, the
limiting distribution at periodic points is not Poissonian (but compound
Poissonian). Here we show that for all non-periodic points the return times are
in the limit Poisson distributed. In the second part we prove that Lai-Sang
Young's Markov Towers have Poisson distributed return times if the correlations
decay for observables that are H\"older continous and $\mathscr{L}^\infty$
bounded.
