In this dissertation, we introduce a new class of spectral time stepping methods for efficient and accurate solutions of ordinary differential equations (ODEs), differential algebraic equations (DAEs), and partial differential equations (PDEs). The methods are based on applying spectral deferred correction techniques as preconditioners to Picard integral collocation formulations, least squares based orthogonal polynomial approximations are computed using Gaussian type quadratures, and spectral integration is used instead of numerically unstable differentiation. For ODE problems, the resulting Krylov deferred correction (KDC) methods solve the preconditioned nonlinear system using Newton-Krylov schemes such as Newton-GMRES method. For PDE systems, method of lines transpose (MoL^T) couples the KDC techniques with fast elliptic equation solvers based on integral equation formulations and fast algorithms. Preliminary numerical results show that the new methods are of arbitrary order of accuracy, extremely stable, and very competitive with existing techniques, particularly when high precision is desired.