  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.
