The theory of modular forms is a subject where the most diverse branches of mathematics come together: complex analysis, algebraic geometry, representation theory, number theory. The last decades have witnessed the emergence of deep connections between modular forms and the p-adic Hodge theory. These relations have been used in many of the recent most remarkable developments in Mathematics such a the proof of Fermat's last theorems. This advanced course will survey this circle of ideas, with a special emphasis on the recent comparison theorems of T. Tsuji and the generalizations, and so in the study of topics such as Serre's conjecture and the recent proof of the full modularity conjecture. The aim of the course is to provide young researchers with the necessary tools to tackle open problems in the subject are, giving them the opportunity to learn the most recent results on p-adic Hodge Theory and also their interpay with modular forms.