This thesis deals with the mathematical field of stochastic nonlinear partial differential equations' analysis. We are interested in parabolic and hyperbolic PDE stochastically perturbed in the Itô sense. We introduce randomness by adding a stochastic integral (Itô integral), which can depend or not on the solution. We thus talk about a multiplicative noise or an additive one. The presence of the random variable does not allow us to apply systematically classical tools of PDE analysis. Our aim is to adapt known techniques of the deterministic setting to nonlinear stochastic PDE analysis by proposing alternative methods. Here are the obtained results: In Chapter I, we investigate on a stochastic perturbation of Barenblatt equations. By using an implicit time discretization, we establish the existence and uniqueness of the solution in the additive case. Thanks to the properties of such a solution, we are able to extend this result to the multiplicative noise using a fixed-point theorem. In Chapter II, we consider a class of stochastic equations of Barenblatt type but in an abstract frame. It is about a generalization of results from Chapter I. In Chapter III, we deal with the study of the Cauchy problem for a stochastic conservation law. We show existence of solution via an artificial viscosity method. The compactness arguments are based on Young measure theory. The uniqueness result is proved by an adaptation of the Kruzhkov doubling variables technique. In Chapter IV, we are interested in the Dirichlet problem for the stochastic conservation law studied in Chapter III. The remarkable point is the use of the Kruzhkov semi-entropies to show the uniqueness of the solution. In Chapter V, we introduce a splitting method to propose a numerical approach of the problem studied in Chapter IV. Then we finish by some simulations of the stochastic Burgers' equation in the one dimensional case.
