In this paper we extend the notion of convergent systems defined by B.P. Demidovich and introduce the notions of uniformly, exponentially convergent and input-to-state convergent systems. Basic (interconnection) properties of such systems are established. Sufficient conditions for input-to-state convergence are presented. For a class of nonlinear systems we design (output) feedback controllers that make the closed-loop system input-to-state convergent. The conditions for such controller design are formulated in terms of LMIs.