In this talk, we discuss a new class of numerical methods for the accurate and efficient integration of time dependent partial differential equations. Unlike traditional method of lines (MoL), the new Krylov deferred correction (KDC) accelerated method of lines transpose (Mol^T) first discretizes the temporal direction using Gaussian type nodes and spectral integration, and the resulting coupled elliptic system is solved iteratively using Newton-Krylov techniques such as Newton-GMRES method, in which each function evaluation is simply one low order time stepping approximation of the error by solving a decoupled system using available fast elliptic equation solvers. Preliminary numerical experiments show that the KDC accelerated MoL^T technique is unconditionally stable, can be spectrally accurate in both temporal and spatial directions, and allows optimal time step sizes in long-time simulations. Numerical experiments for parabolic type equations including the Schrodinger equation will be discussed.