We present a framework for compatible spatial discretizations and time integrators that incorporates properties of the combined space-time integrals of a given PDE. Demonstrations for model equations will be provided, ranging from hyperbolic conservation laws, such as Lagrangian advection and Maxwell’s equations, and parabolic systems like heat transfer with phase change. The approach is based on a “cut-cell” discretization that tracks interfaces and jump conditions of the PDE along an approximate space-time boundary. It can also be extended to include moving meshes, and even mesh discontinuities in space-time, which otherwise would introduce large errors and overly-restrictive time steps for stability with typical approaches. We'll demonstrate that in comparison with a generic method-of-lines discretization, better accuracy and stability properties can be achieved when considering design of space- and time-discretizations together.