We present an introduction to the main concepts and proofs of Malliavin calculus. We define the notions of gradient and divergence at the same time for the Brownian motion and for fractional Brownian motions as they both come naturally from the Wiener space structure which is similar for these processes. We then show the basic similarities and difference with the stochastic calculus of variations for Poisson process. A new framework is then introduced for family of independent random variables. The Dirichlet form for Wiener and Poisson spaces can be recovered by limiting procedures starting from this newly defined structure. We then show that how these elements interplay in the Stein-Malliavin-Dirichlet method.