Appears in collection : 8ème Séminaire Itzykson : "Les observables parafermioniques et la physique statistique en 2D"

During the last decade, a number of results on conformal invariance of the critical 2D nearest-neighbor Ising model appeared, both for the small mesh size limits of correlation functions (fermions, energy densities, spins, disorders etc) and interfaces (aka domain walls) or loop ensembles arising in the model. We survey these results, which are based on discrete analysis techniques, notably on convergence theorems for solutions to some special boundary value problems for discrete holomorphic functions on rhombic lattices, and discuss the universality of the model with respect to an underlying planar graph.