An intrinsic feature of nearly all internal interfaces in crystalline systems (homo- and hetero-phase) is the presence of disconnections, namely topological line defects constrained to the interface that have both step and dislocation character. We demonstrate that elastic interactions between disconnections strongly affect the morphology and motion of interfaces, allowing for understanding and reconciling diverse key experiments. In particular, these elastic interactions strongly modify equilibrium interface morphologies compared with those solely determined by anisotropic surface energy, and affect the kinetics of migrating interfaces. They are also found to lead to a thermodynamic, first-order, finite-temperature, faceting–defaceting transition. We demonstrate these phenomena through numerical simulations based upon a general, continuum disconnection-based model for interface thermodynamics and kinetics applied to embedded particles/grains, steady-state interface migration geometries, and nominally flat interfaces.