This review article deals with the regularization of the boundary element formulations for solution of boundary value problems of continuum mechanics. These formulations may be singular owing to the use of two-point singular fundamental solutions. When the physical interpretation is irrelevant for this topic of computational mechanics, we consider various mechanical problems simultaneously within particular sections selected according to the main topic. In spite of such a structure of the paper, applications of the regularization techniques to many mechanical problems are described. There are distinguished two main groups of regularization techniques according to their application to singular formulations either before or after the discretization. Further subclassification of each group is made with respect to basic principles employed in individual regularization techniques. This paper summarizes the substances of the regularization procedures which are illustrated on the boundary element formulation for a scalar potential field. We discuss the regularizations of both the strongly singular and hypersingular integrals, occurring in the boundary integral equations, as well as those of nearly singular and nearly hypersingular integrals arising when the source point is near the integration element (as compared to its size) but not on this element. The possible dimensional inconsistency (or scale dependence of results) of the regularization after discretization is pointed out. Finally, we discuss the numerical approximations in various boundary element formulations, as well as the implementations of solutions of some problems for which derivative boundary integral equations are required.