  This paper presents the first analysis of a space--time hybridizable
discontinuous Galerkin method for the advection--diffusion problem on
time-dependent domains. The analysis is based on non-standard local trace and
inverse inequalities that are anisotropic in the spatial and time steps. We
prove well-posedness of the discrete problem and provide a priori error
estimates in a mesh-dependent norm. Convergence theory is validated by a
numerical example solving the advection--diffusion problem on a time-dependent
domain for approximations of various polynomial degree.
