  In this paper, we propose new geometrically unfitted space-time Finite
Element methods for partial differential equations posed on moving domains of
higher order accuracy in space and time. As a model problem, the
convection-diffusion problem on a moving domain is studied. For geometrically
higher order accuracy, we apply a parametric mapping on a background space-time
tensor-product mesh. Concerning discretisation in time, we consider
discontinuous Galerkin, as well as related continuous (Petrov-)Galerkin and
Galerkin collocation methods. For stabilisation with respect to bad cut
configurations and as an extension mechanism that is required for the latter
two schemes, a ghost penalty stabilisation is employed. The article puts an
emphasis on the techniques that allow to achieve a robust but higher order
geometry handling for smooth domains. We investigate the computational
properties of the respective methods in a series of numerical experiments.
These include studies in different dimensions for different polynomial degrees
in space and time, validating the higher order accuracy in both variables.
