We consider fully nonlinear elliptic equations on complex manifolds which depend on the gradient in some nontrivial ways. Some of these equations arise from interesting problems in complex geometry, such as a conjecture by Gauduchon which is a natural generalization of Calabi conjecture to the Hermitian setting, and finding balanced metrics on Hermitian manifolds. We shall discuss difficulties in solving such equations and present recent results in our attempt to overcome these difficulties.  Our goal is to establish some general existence results which we hope will find useful applications in complex geometry in the near future. We'll explain how our results provide a proof to the Gauduchon conjecture building on previous work of Tossati-Weinkove and others. The talk is based on joint work with Xiaolan Nie, Chunhui Qiu and Rirong Ruan.