In Chap. 2 , the coupling constructions are described for discrete finite-values random variables, random variables with absolutely continuous distributions, and general random variables taking values in measurable spaces. We give the explicit expressions for maximal coupling probability for random vectors with prescribed marginal distributions of components, the explicit form of the two-dimensional distribution with maximal coupling probability, and stochastic representation for a random vector with maximal coupling probability, as well as give formulas connecting the maximal coupling probability with the variational distance for the corresponding marginal distributions and an explicit formula for this variational distance for absolutely continuous distributions it in terms of the corresponding marginal probability density functions. We also describe the structure of the family of two-dimensional distribution with conditional maximal coupling probability, and give some results concerning the approximation of maximal coupling probability for general and discrete random variables.