Realistic quantum mechanical systems are influenced through the coupling to an environment containing a large number of mostly uncontrollable degrees of freedom. This unavoidable interaction of an open quantum systems with its environment leads to the mechanisms of dissipation and damping, and to a strong and often rapid loss of quantum coherence. The talk begins with a brief introduction into the standard theory of quantum mechanical relaxation which is based on the Markov approximation and on the concepts of completely positive dynamical semigroups and of quantum master equations in Lindblad form. Many examples for this approach are known from quantum optics, decoherence theory, quantum Brownian motion and quantum measurement and control theory. However, strong couplings or interactions with low-temperature reservoirs generally lead to large system-environment correlations which result in long memory times and in a failure of the Markov approximation. To describe the basic features of the non-Markovian quantum dynamics of open systems we develop several new methods as, for example, the technique of correlated projection superoperators [1] and the concept of quantum semi-Markov processes [2]. A number of examples and applications to structured and finite reservoirs [3], to electron spin dynamics in quantum dots [4], and to the problem of quantum transport in nano-structures [5] will be discussed. [1] H. P. Breuer, Phys. Rev. A 75, 022103 (2007). [2] H. P. Breuer and B. Vacchini, Phys. Rev. Lett. 101, 140402 (2008). [3] H. P. Breuer, J. Gemmer and M. Michel, Phys. Rev. E73, 016139 (2006). [4] E. Ferraro, H. P. Breuer, A. Napoli, M. A. Jivulescu, and A. Messina, Phys. Rev. B78, 064309 (2008). [5] R. Steinigeweg, H. P. Breuer and J. Gemmer, Phys. Rev. Lett. 99, 150601 (2007).