A convergent system is a system of differential equations which has a unique solution satisfying the following two properties:
1. this solution is defined and bounded on the whole time scale (from minus- to plus infinity),
2. it is globally asymptotically stable.
It appears that (nonlinear) convergent systems have certain properties which significantly simplify their analysis relative to analysis of general nonlinear systems. Also the notion of convergent systems appears to be very useful in many problems related to control of nonlinear systems such as the problem of observer design, the controlled synchronization problem, and the output regulation problem. In this presentation we give definitions of convergent systems, discuss their properties and present conditions under which a system is convergent. Several examples illustrating the application of convergent systems to control are given.