I will begin by presenting earlier joint work with E. M. Stein concerning div-curl type inequalities for the exterior derivative complex and its adjoint in Euclidean space R^n. I will then offer various, higher-order generalizations of the notion of exterior derivative that stem from recent work of Bourgain-Brezis and Van Schaftingen, and discuss related div-curl type estimates for such operators. Part of this work is joint with A. Raich.