Many practical applications of compressible flows involve time-varying geometries with large deformations including mixers with rotating objects, pitching tandem airfoils and deploying spoilers on airfoils. In this talk, we present a high-order accurate discontinuous Galerkin (DG) method solving two-dimensional compressible flow with fully unstructured space-time meshes. The discretization of our space-time framework is based on a nodal DG 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 Newton-Krylov solver. The spatial meshes are produced by some mesh moving techniques with element connectivity updates, and the corresponding space-time elements are produced directly based on these local operations. Finally, we present an efficient algorithm to generate the globally conforming tetrahedral meshes based on those local space-time elements. The talk is based on joint work with Per-Olof Persson (UC Berkeley).