In this book we study Hilbert schemes of zero-dimensional subschemes of smooth varieties and several related parameter varieties of interest in enumerative geometry. The main aim here is to describe their cohomology and Chow rings. Some enumerative applications are also given. The Weil conjectures are used to compute the Betti numbers of many of the varieties considered, thus also illustrating how this powerful tool can be applied. The book is essentially self-contained, assuming only a basic knowledge of algebraic geometry; it is intended both for graduate students and research mathematicians interested in Hilbert schemes, enumertive geometry and moduli spaces.