This paper gives a survey of our recent work on spacetime spectral methods for PDEs and an exposition of some current progress. 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 PDEs defined on regular geometry. The method is extremely robust across all types of PDEs (dispersive, diffusive, presence of advection terms with all standard boundary conditions). The main drawback of space-time spectral methods is that time marching is no longer feasible – all unknowns in both space and time must be solved for simultaneously.