A new formalism [1,2] for calculating exact steady-state non-linear multi-mode lasing states for complex resonators is developed and applied to conventional edge-emitting lasers and to lasers with chaotic or random cavities. The theory solves a long-standing problem in lasing theory: how to describe the multi-mode lasing states of an open cavity. Moreover it includes the effects of mode competition and spatial hole-burning to all orders within the approximation of stationary inversion. Lasing modes are expanded in terms of sets of biorthogonal "constant flux" (CF) states and satisfy a self-consistent equation. For high finesse cavities each lasing mode is proportional to one CF state which inside the cavity behaves like a linear resonance; for low finesse as in a random laser, novel composite modes are predicted which do not correspond to any passive cavity resonance.