[en] Currently interesting higher-dimensional theories have degenerate ground state spacetimes of the form Minkowski spacetime (M_d) times a compact space (B_n). The degeneracy arises because neither the size nor the shape of B_n is determined by the classical field equations. The size and shape parameters define a moduli space and we show that all d-dimensional Friedmann-Robertson-Walker cosmologies correspond to geodesic motion in this moduli space. If the universe is open and expands forever, this motion is damped and at late times the D = (d + n)-dimensional universe tends to M_d . B_n where the metric on B_n is time independent. Inflation is not possible in scale invariant theories, but we indicate how a breakdown in scale invariance at the quantum level could lead to inflation. We also discuss some bizarre aspects of the global structure of some of these D-dimensional cosmologies. (orig.)