We first plan to focus on discrete models defined on lattices and discuss: - The phase transition for percolation and for the Ising model. - The asymptotic conformal invariance of critical percolation on the triangular planar lattice (“Smirnov’s Theorem”). - Wilson’s algorithm to construct uniform spanning trees. - Some properties of the discrete Gaussian Free Field. In the final part of the course, we will discuss some features of random structures defined in the continuum: - The definition of the Gaussian Free Field as a random generalized function and some of its properties. - How conformal invariance enables to characterize and construct natural random planar fractal structures.