Master equations are partial differential equations for measure-dependent unknowns, and are introduced to describe asymptotic equilibrium of large scale mean-field interacting systems, especially in stochastic games and control. In this paper we introduce new semilinear master equations whose unknowns are functionals of both paths and path measures. They include state-dependent master equations, path-dependent partial differential equations (PPDEs), history information dependent equations and time inconsistent (e.g. time-delayed) equations, which naturally arise in applications (e.g. option pricing, risk control). We give a classical solution to the master equation by introducing a new notation called strong vertical derivative (SVD) for path-dependent functionals, inspired by Dupire’s vertical derivative, and applying forward-backward stochastic system argument. This talk is based on a joint work with Shanjian Tang.