  We consider a class of fully nonlinear second order elliptic equations on
Hermitian manifolds closely related to the general notion of
$\bfG$-plurisubharmonicity of Harvey-Lawson and an equation treated by
Sz\'ekelyhidi-Tosatti-Weinkove in the proof of Gauduchon conjecture. Under
fairly general assumptions we derive interior estimates and establish the
existence of smooth solutions for the Dirichlet problem as well as for
equations on closed manifolds.
