Time is, figuratively and literally, becoming the new dimension for crystalline matter. As such, rapid recent progress on time-varying media gave rise to the notion of temporal and spatiotemporal crystals. Fundamentally rethinking the role of time, which, in contrast to space exhibits a unique unidirectionality often referred to as the arrow of time, promises a new dimension also for topological physics. Here, we enter the new realm of time and spacetime topology: Firstly, we implement a time-topological time interface state. Secondly, we propose and observe a spacetime-topological event and demonstrate unique features like its limited collapse under disorder and causality-suppressed coupling. The new paradigms of time and spacetime topology unveil a distinctive role of causality and non-Hermiticity in topology and pave the way towards topological spatiotemporal wave control with unique robustness. 