© 2015 Elsevier Ltd. We present a high-order accurate space-time discontinuous Galerkin method for solving two-dimensional compressible flow problems on fully unstructured space-time meshes. The discretization is based on a nodal formulation, with appropriate numerical fluxes for the first and the second-order terms, respectively. The scheme is implicit, and we solve the resulting non-linear systems using a parallel Newton-Krylov solver. The meshes are produced by a mesh moving technique with element connectivity updates, and the corresponding space-time elements are produced directly based on these local operations. To obtain globally conforming tetrahedral meshes, we first derive the required conditions on a prism boundary mesh to allow for a valid local triangulation. Next, we present an efficient algorithm for finding a global mesh that satisfies these conditions. We also show how to add and remove mesh nodes, again using local constructs for the space-time mesh. Our method is demonstrated on a number of test problems, showing the high-order accuracy for model problems, and the ability to solve flow problems on domains with complex large deformations.