These notes are an extended version of a set of lectures given at ”MONICA: MONomial Ideals, Computations and Applications”, at the CIEM, Castro Urdiales (Cantabria, Spain) in July 2011. The goal of these lectures is to give an overview of some results that have been developed in recent years about the structure of local cohomology modules supported on a monomial ideal. We will highlight the interplay of multi-graded commutative algebra, combinatorics and D-modules theory that allow us to give different points of view to this subject. We will try to preserve the informal character of the lectures, so very few complete proofs are included