are false. The local and global properties of surfaces are significantly different. By proving assertion 1) A.V. Pogorelov completed the solution of the problem open for more than a century. The first result in this direction was obtained by Cauchy for closed convex polyhedra in 1813. The theorems proved by Pogorelov formed the basis for his nonlinear theory of thin shells. This theory is concerned with those elastic states of the shell which differ significantly comparing to the original form. Under such deformations, the middle surface of a thin shell undergoes bending with preservation of the metric. This makes it possible, by using theorems proved by Pogorelov for convex surfaces, to investigate loss of stability and the over critical elastic state of convex shells under a given strain. Such shells are the most common elements of modern designs. Results 1) and 2) were generalized for regular surfaces in a Riemannian space. In addition, the Weyl problem for Riemannian space was solved: it was proved that a regular metric of Gaussian curvature greater than some constant c on a two-dimensional sphere can be isometrically immersed into a complete three-dimensional Riemannian space of curvature <c in a form of a regular surface. Studying the methods developed in the proof of this result, the Abel Prize laureate M. Gromov introduced the concept of pseudoholomorphic curves, which are the main tool in modern symplectic geometry. A closed convex hypersurface is uniquely defined not only by the metric but also by the Gaussian curvature as a function of unit normals. Moreover, the hypersurface is uniquely determined up to a parallel transport. This was proved by G. Minkowski. But is the hypersurface regular under the condition that the Gaussian curvature K(n) is a regular function of a unit normal? Pogorelov proved that if positive function K(n) belongs to the class Сk, k≥3, then the support function will be of regularity class Сk+1,v, 0<v<1. The hardest part of the proof of the theorem was to obtain a priori estimates for the derivatives of the support function of a hypersurface up to third order inclusively. Pogorelov's method of a priori estimates was used by S.-T. Yau to obtain a priori estimates for solutions of complex Monge-Ampere equations. This was the main step in the proof of the existence of Calabi-Yao manifolds, which play an important role in theoretical physics. A Monge-Ampère equation has the form A priori estimates