The purpose of this book is to present results on the subject of weak convergence in function spaces to study invariance principles in statistical applications to dependent random variables, U-statistics, censor data analysis. Different techniques, formerly available only in a broad range of literature, are for the first time presented here in a self-contained fashion. Contents:Weak convergence of stochastic processes Weak convergence in metric spaces Weak convergence on C[0, 1] and D[0, infinity)Central limit theorem for semi-martingales and applications Central limit theorems for dependent random variables Empirical process Bibliography