We clarify the properties of the behavior of classical cosmological perturbations when the Universe experiences a bounce. This is done in the simplest possible case for which gravity is described by general relativity and the matter content has a single component, namely, a scalar field in a closed geometry. We show in particular that the spectrum of scalar perturbations can be affected by the bounce in a way that may depend on the wave number, even in the large scale limit. This may have important implications for string motivated models of the early Universe.