  We introduce a space-time Trefftz discontinuous Galerkin method for the
first-order transient acoustic wave equations in arbitrary space dimensions,
extending the one dimensional scheme of Kretzschmar et al. (2016, IMA J. Numer.
Anal., 36, 1599-1635). Test and trial discrete functions are space-time
piecewise polynomial solutions of the wave equations. We prove well-posedness
and a priori error bounds in both skeleton-based and mesh-independent norms.
The space-time formulation corresponds to an implicit time-stepping scheme, if
posed on meshes partitioned in time slabs, or to an explicit scheme, if posed
on "tent-pitched" meshes. We describe two Trefftz polynomial discrete spaces,
introduce bases for them and prove optimal, high-order $h$-convergence bounds.
