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In Chap. 1 the concepts of the nuclear magnetic relaxation times (rates) were introduced on the basis of phenomenological considerations, as exemplified by the theory of Bloch. However, this approach can not clarify the nature of relaxation processes and quantify the relaxation times of various nuclei in different environments. The answers to these questions are given by quantum theory, which allows the calculation of the probability of relaxation transitions. In this chapter the problems of the relaxation in two-and multi-spin systems will be considered using the theory of time-dependent perturbations. Probability Relaxation TransitionsThe interaction of nuclei with the environment (neighboring nuclei, unpaired electrons and etc.) can lead, firstly, to non-equidistant energy levels and, consequently, the splitting of the spectrum of the nuclear magnetic resonance, secondly, to non-radiative transitions of nuclei between different energy states (exchange of energy between different degrees of freedom inside a substance). These non-radiative transitions have usually chaotic character and are a fundamental mechanism of the achievement of an equilibrium (Boltzmann) distribution of nuclei among energy states. These processes, which are the main object of consideration in this chapter, are called as relaxational. In general, the Hamiltonian of the nuclei system can be written aswhere H 0 describes the interaction of nuclei with a constant magnetic field B 0 , H describes their interaction with the environment. In the investigations of nuclear magnetic resonance in the majority of cases one deals with weakly interacting nuclei. Therefore, the Hamiltonians H 0 and H contain terms, taking into account the interaction of an isolated nucleus or a small number of nuclei with the external field B 0

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