Abstract: 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 Szekelyhidi-Tosatti-Weinkove in the proof of Gauduchon conjecture. Such equations also appear in Chern-Ricci type flows and conformal deformations of Chern-Ricci forms. 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. The talk is based on joint work with Mathew George and Chunhui Qiu.