The notion of an elliptic partial differential equation (PDE) goes back at least to 1908, when it appeared in a paper J. Hadamard. In this talk we present a recently discovered structural condition, called $p$-ellipticity, which generalizes classical ellipticity. It was co-discovered independently by Carbonaro and Dragicevic on one hand, and Pipher and myself on the other, and plays a fundamental role in many seemingly unrelated aspects of the $L^p$ theory of elliptic complex-valued PDE. So far, $p$-ellipticity has proven to be the key condition for: (i) convexity of power functions (Bellman functions) (ii) dimension-free bilinear embeddings, (iii) $L^p$-contractivity and boundedness of semigroups $(P_t^A)_{t>0}$ associated with elliptic operators, (iv) holomorphic functional calculus, (v) multilinear analysis, (vi) regularity theory of elliptic PDE with complex coefficients. During the talk, I will describe my contribution to this development, in particular to (vi).