We introduce a new formalism of higher-dimensional timed automata, based on Pratt and van Glabbeek’s higher-dimensional automata and Alur and Dill’s timed automata. We prove that their reachability is PSPACE-complete and can be decided using zone-based algorithms. We also extend the setting to higher-dimensional hybrid automata.The interest of our formalism is in modeling systems which exhibit both real-time behavior and concurrency. Other existing formalisms for real-time modeling identify concurrency and interleaving, which, as we shall argue, is problematic.