We present a new unified theory of critical finite-size scaling for lattice statistical mechanical models with periodic boundary conditions above the upper critical dimension. Our theory is based on recent mathematically rigorous results for linear and branched polymers, multi-component spin systems, and percolation. Both short-range and long-range interactions are included. The universal finite-size scaling is inherited from the scaling of the system unwrapped to the infinite lattice. We also present conjectures for universal scaling profiles for the susceptibility and two-point function plateau in a critical window. For free boundary conditions, the universal scaling has been proven to apply at a pseudocritical point for hierarchical spins, and we conjecture that this holds generally.