Many important phenomena are modelled by time-dependent partial differential equations whose accurate solution is challenging and costly. Numerical algorithms for these problems must satisfy particular stability and accuracy constraints while marching as rapidly as possible. I will introduce some examples of valuable but costly time-dependent wave equation computations. Then I will give an introduction to key stability concepts in time discretization of PDEs, followed by a description of my recent work aimed at answering questions like • How can we design accurate, stable numerical methods that march faster in time? • Can we find methods that ensure positivity of solutions? • What are the theoretical limits on achievable efficiency of time discretizations for PDEs?