One of the most interesting questions of the course of geometry was considered in the work - the question of the equivalence and equivalence of geometric figures, which was raised at the end of the 19th century. the great mathematician D. Hilbert (he formulated it in the form of a third problem). All kinds of tasks are considered that clarify the essence of the problem under consideration. We limited our work to considering only flat geometric shapes, although this problem is considered in geometry and in the case of volumetric geometric shapes.