We present a new language semantics for real-time concurrency. Its operational models are higher-dimensional timed automata (HDTAs), a generalization of both higher-dimensional automata and timed automata. In real-time concurrent systems, both concurrency of events and timing and duration of events are of interest. Thus, HDTAs combine the non-interleaving concurrency model of higher-dimensional automata with the real-time modeling, using clocks, of timed automata. We define languages of HDTAs as sets of interval-timed pomsets with interfaces. We show that language inclusion of HDTAs is undecidable. On the other hand, using a region construction we can show that untimings of HDTA languages have enough regularity so that untimed language inclusion is decidable. On a more practical note, we give new insights on when practical applications, like checking reachability, might benefit from using HDTAs instead of classical timed automata.