Abstract: Space-time parallel methods, also known more recently under the name PinT (Parallel in Time) methods, have a long history, but they have received a lot of attention over the past two decades. This is driven by the parallel hardware architectures that have now millions of cores, leading to saturation when parallelizing in space only. Parallelizing also the time direction is tempting, but this is very different from the space direction, because evolution problems satisfy a causality principle: the future is dependent on the past, and not the other way round. I will show in my presentation that successful strategies for PinT methods depend strongly on the nature of the evolution problem. For hyperbolic problems, effective PinT methods are Domain Decomposition methods of Waveform Relaxation type, culminating in Unmapped Tent Pitching methods, ParaDiag methods, and also direct time parallel methods like ParaExp. Most of these methods can also be very effectively used for parabolic problems, but for such problems there are also highly successful multilevel methods, like Parareal and its variants. The currently best ones are however space-time multigrid methods. All these multilevel methods struggle however when applied to hyperbolic problems.