I give a pedagogical introduction to Brownian motion, stochastic calculus introduced by Ito in the fifties, following the elementary (at least not too technical) approach by Föllmer 1981. Based on this, I develop the connection with linear and semi-linear parabolic PDEs. Then, I provide and analyze some Monte Carlo methods to approximate the solution to these PDEs This course is aimed at master students, PhD students and researchers interesting in the connection of stochastic processes with PDEs and their numerical counterpart. The reader is supposed to be familiar with basic concepts of probability (say first chapters of the book "Probability essentials" by Jacod and Protter 2003), but no a priori knowledge on martingales and stochastic processes is required.