In this talk we will discuss the fundamental aspects of the theory of compensated compactness in the general A-free framework developed by Murat and Tartar, under the assumption that A has constant rank. We prove sharp weak continuity results for the compensated compactness quantities and we show that they are precisely the nonlinear quantities with Hardy space integrability, thus proposing an answer to a question raised by Coifman-Lions-Meyer-Semmes. This gives a link between compensated compactness and compensated regularity.