This paper provides an overview of two classes of numerical schemes for solving dynamic elasticity and plasticity problems based on incompatible \(3D\) finite elements developed and implemented by the authors. The review includes a theoretical justification, a description of the method for constructing numerical schemes and their properties and advantages in solving nonstationary problems. A feature of the first class of rare mesh FEM schemes is the arrangement of the calculated 4-node finite elements in the form of tetrahedrons with regular intervals inside a grid of hexahedral cells. Otherwise, this can be interpreted as a scheme of a 4-node hexahedral finite element. Due to the successful mutual arrangement of the finite elements, these schemes have a number of advantages over traditional FEM schemes with continuous filling of the computational domain by finite elements. This is high efficiency, as well as the absence of two undesirable effects–shear locking and hourglass instability. In terms of convergence and stability, this scheme is not inferior to traditional schemes, in particular, the well-known Wilkins difference scheme. This is confirmed by theoretical analysis and the results of solving a large number of elasticity and plasticity problems. As a development of the idea of rare mesh schemes, a class of moment finite elements is considered, based on the projection of rare mesh schemes onto grids of lower dimension. An 8-node finite element in the form of a hexahedron with constant moment components of the stress tensor within the element, constructed on the basis of this approach, retains all the best properties of rare mesh schemes. The results of solving a number of dynamic and static elasticity and plasticity problems are presented.