The notion of Hilbert Function is central in Commutative Algebra and it has important applications in Algebraic Geometry, Combinatorics, Singularity Theory and Computational Algebra.

In this thesis we shall deal with different aspects of the theory of Hilbert functions by presenting our contribution concerning three problems of great interest through the last decades in this area of dynamic mathematical activity. The first part of the thesis concerns the study of the Hilbert function of standard graded algebras. In his famous paper "Uber die Theorie der algebraischen Formen" (see [28]) pub- ¨ lished more than a century ago, Hilbert proved that a graded module M over a polynomial ring has a finite graded resolution and he concluded from this fact that its Hilbert function is of polynomial type. The Hilbert function of the homogeneous coordinate ring of a projective variety V , which classically was called the postulation of V , is a rich source of discrete invariants of V and its embedding. The dimension, the degree and the arithmetic genus of V can be immediately read from the generating function of the Hilbert function