In this paper, a general asymptotic theory of synchronization of the chaotic oscillations for non-identical dissipative-coupled dynamical systems is proposed. The theory is based on the general definition of synchronization and on the method of integral manifolds. A number of different cases of non-identical dynamical systems and their couplings when the synchronization is asymptotically close to the identical one have been considered. This theory is mutually valid for the master and slave in synchronization of dynamical systems including the systems with slowly varying parameters. Theoretical findings are supported by the results of numerical simulation.