In this talk we discuss the roles of concavity, symmetry and subsolutions in the study of fully nonlinear PDEs, especially those on real or complex manifolds with connection to geometric problems. We shall report some of our results along the line, which give the optimal conditions for the existence of classical solutions, either of the Dirichlet problem, or of equations on closed manifolds. If time permits, we shall also discuss the possibility to weaken or extend these conditions, and a class of equations involving differential forms of higher rank, more specifically real (p, p) forms for p > 1 on complex manifolds. Part of the talk is based on joint work with my student Mathew George.