The courses are introductory to carry research activity in contemporary trends in Hodge Theory and p-adic Hodge Theory given by main actors in each activity: Hodge structures, Hodge-Tate structures, De Rham Cohomology and Betti Cohomology, Torelli type theorems, Abelian Varieties. p-adic numbers, p-adic Galois representations, absolute Galois group of a p-adic field, continuos representations, Fontaine rings and admisible representations, Hodge-Tate representations, étale cohomology as a p-adic Galois representation, Tate module of an elliptic curve, ring of periods, De Rham representations, good reduction and crystalline representations.