  We establish existence and uniqueness for a wide class of Markovian systems
of backward stochastic differential equations (BSDE) with quadratic
nonlinearities. This class is characterized by an abstract structural
assumption on the generator, an a-priori local-boundedness property, and a
locally-H\"older-continuous terminal condition. We present easily verifiable
sufficient conditions for these assumptions and treat several applications,
including stochastic equilibria in incomplete financial markets, stochastic
differential games, and martingales on Riemannian manifolds.
