  We introduce languages of higher-dimensional automata (HDAs) and develop some
of their properties. To this end, we define a new category of precubical sets,
uniquely naturally isomorphic to the standard one, and introduce a notion of
event consistency. HDAs are then finite, labeled, event-consistent precubical
sets with distinguished subsets of initial and accepting cells. Their languages
are sets of interval orders closed under subsumption; as a major technical step
we expose a bijection between interval orders and a subclass of HDAs. We show
that any finite subsumption-closed set of interval orders is the language of an
HDA, that languages of HDAs are closed under binary unions and parallel
composition, and that bisimilarity implies language equivalence.
