  We derive a priori estimates for solutions of a general class of fully
non-linear equations on compact Hermitian manifolds. Our method is based on
ideas that have been used for different specific equations, such as the complex
Monge-Amp\`ere, Hessian and inverse Hessian equations. As an application we
solve a class of Hessian quotient equations on K\"ahler manifolds assuming the
existence of a suitable subsolution. The method also applies to analogous
equations on compact Riemannian manifolds.
