We will explain the basic concepts of Homological Algebra (Cech cohomology, injective/projective resolutions, derived functors,) and show why, useful as they are, they have shortcomings for a general theory of cohomology. Time allowing, we'll give a hint of why derived categories make up for those defficiencies.

(**) Artur de Araujo is a PhD student for the Joint PhD Program in Mathematics UC|UP working at the University of Porto in the area of Geometry and Topology under the supervision of professor Peter Gothen.