0.1 The partial differential equation (1) (Au)(x) = L aa(x)(Dau)(x) = f(x) m lal9 is called elliptic on a set G, provided that the principal symbol a2m(X,¿) = L aa(x)¿a lal=2m of the operator A is invertible on G X (~n 0); A is called elliptic on G, too. This definition works for systems of equations, for classical pseudo differential operators ("pdo), and for operators on a manifold n. Let us recall some facts concerning elliptic operators. 1 If n is closed, then for any s E ~ , is Fredholm and the following a priori estimate holds (2) 1 2 Introduction If m > 0 and A : C=(O; C') -+ L (0; C') is formally self - adjoint 2 with respect to a smooth positive density, then the closure Ao of A is a self - adjoint operator with discrete spectrum and for the distribu­ tion functions of the positive and negative eigenvalues (counted with multiplicity) of Ao one has the following Weyl formula: as t -+ 00, (3) n 2m = t / II N±(1,a2m(x,e))dxde T·OO (on the right hand side, N±(t,a2m(x,e))are the distribution functions of the matrix a2m(X,e) : C' -+ CU).

0. Introduction. 1. General Calculus of Pseudodifferential Operators. 2. Model Classes of Degenerate Elliptic Differential Operators. 3. General Classes of Degenerate Elliptic Differential Operators. 4. Degenerate Elliptic Operators in Non--Power--Like Degeneration Case. 5. Lp-Theory for Degenerate Elliptic Operators. 6. Coerciveness of Degenerate Quadratic Forms. 7. Some Classes of Hypoelliptic Pseudodifferential Operators on a Closed Manifold. 8. Algebra of Boundary Value Problems for Class of Pseudodifferential Operators which Change Order on the Boundary. 9. General Schemes of Investigation of Spectral Asymptotics for Degenerate Elliptic Equations. 10. Spectral Asymptotics of Degenerate Elliptic Operators. 11. Spectral Asymptotics of Hypoelliptic Operators with Multiple Characteristics. A Brief Review of the Bibliography. Bibliography. Index of Notations. Index.