  We establish the general equivalence between rare event process for arbitrary
continuous functions whose maximal values are achieved on non-trivial sets, and
the entry times distribution for arbitrary measure zero sets. We then use it to
show that the for differentiable maps on a compact Riemannian manifold that can
be modeled by Young's towers, the rare event process and the limiting entry
times distribution both converge to compound Poisson distributions. A similar
result is also obtained on Gibbs-Markov systems, for both cylinders and open
sets. We also give explicit expressions for the parameters of the limiting
distribution, and a simple criterion for the limiting distribution to be
Poisson. This can be applied to a large family of continuous observables that
achieve their maximum on a non-trivial set with zero measure.
