Stability and convergence analysis for fully discrete pseudo spectral numerical schemes to nonlinear PDEs are presented in this talk, such as viscous Burgers' equation and incompressible Navier-Stokes equations. Related applications to incompressible Euler equation and quasi-geostrophic equation will also be addressed, in both 2-D and 3-D, for smooth and vortex sheet initial data. In addition, high order time stepping schemes, including Adams Bashforth-Adams Moulton multi-step schemes up to fourth order accuracy and high order explicit SSP schemes, will be explored in detail. Unconditional stability is established for the implicit time stepping algorithms.