This paper provides a review of the use of composite grids in the finite difference method (FDM) for solving various problems in mathematical physics. In the first part of the paper, the author describes a method for the generation of grids adapting to curved boundaries of plane domains. Then this method is used to construct orthogonal curvilinear grids near the boundary, or near interfaces, as one component of a composite grid. In the elliptic case, the composite grid FDM is defined as a discrete version of the alternating Schwarz method in connection with the method of fictitious domains. Generalizations to singularly perturbed problems, convection-diffusion problems, problems with free boundaries, parabolic and hyperbolic problems are discussed. A list of 43 references, mainly to Russian authors, completes the paper.