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).