We review representation results of the random solutions by so-called "generalized polynomial chaos" (gpc) expansions in countably many variables. We present recent mathematical results on regularity of such solutions as well as computational approaches for the adaptive numerical Galerkin and Collocation approximations of the infinite dimensional parametric, deterministic solution. A key principle are new sparsity estimates of gpc expansions of the parametric solution. We present such estimates for elliptic, parabolic and hyperbolic problems with random coefficients, as well as eigenvalue problems. We compare the possible convergence rates with the best convergence results on Monte Carlo Finite Element Methods (MCFEM) and on MLMCFEM.