Spectral discretization of an elliptic PDE with a smooth solution yields a numerical solution that converges spectrally to the true solution. Classical spectral methods for time-dependent PDEs use low-order finite difference discretization of the time derivative, and spectral discretization of the spatial derivatives, creating a large imbalance in the temporal and spatial discretization errors. Space-time spectral methods address this deficiency by employing spectral discretization in time as well. Two main advantages of space-time spectral methods include full spectral convergence and ease of implementation for linear PDEs defined on regular geometry. The method is also extremely robust, working well regardless of the type and order of PDE (dispersive, diffusive, presence of advection terms, all standard boundary conditions). The main drawback of space-time spectral methods is that time marching is no longer feasible - the numerical solution must be solved for all unknowns in both space and time simultaneously. This talk gives a brief survey of past work done by my group on space-time spectral methods, and reports on recent progress on these methods for Navier-Stokes and MHD equations, and PDEs defined on irregular domains. This is joint work with A. Kaur, S. Nataj and C. P. Wilegoda Liyanage.