Two-sided estimates for polynomials related to Newton's polygon and their application to studying local properties of partial differential operators in two variables.- Newton's polygon of a polynomial in two variables.- Polynomials admitting of two-sided estimates.- N Quasi-elliptic polynomials in two variables.- N Quasi-elliptic differential operators.- Appendix to - 2. Parabolic operators associated with Newton's polygon.- Polynomials correct in Petrovski?'s sense.- Two-sided estimates for polynomials in two variables satisfying Petrovski?'s condition. N-parabolic polynomials.- Cauchy's problem for N-stable correct and N-parabolic differential operators in the case of one spatial variable.- Stable-correct and parabolic polynomials in several variables.- Cauchy's problem for stable-correct differential operators with variable coefficients.- 3. Dominantly correct operators.- Strictly hyperbolic operators.- Dominantly correct polynomials in two variables.- Dominantly correct differential operators with variable coefficients (the case of two variables).- Dominantly correct polynomials and the corresponding differential operators (the case of several spatial variables).- 4. Operators of principal type associated with Newton's polygon.- Introduction. Operators of principal and quasi-principal type.- Polynomials of N-principal type.- The main L2 estimate for operators of N-principal type.- Appendix to - Local solvability of differential operators of N-principal type.- Appendix to - 5. Two-sided estimates in several variables relating to Newton's polyhedra.- Estimates for polynomials in ?n relating to Newton's polyhedra.- Two-sided estimates in some regions in ?n relating to Newton's polyhedron. Special classes of polynomials and differential operators in several variables.- 6. Operators of principal type associated with Newton's polyhedron.- Polynomials of N-principal type.- Estimates for polynomials of N-principal type in regions of special form.- The covering of ?n by special regions associated with Newton's polyhedron.- Differential operators of ?n-principal type with variable coefficients.- Appendix to - 7. The method of energy estimates in Cauchy's problem Introduction. The functional scheme of the proof of the solvability of Cauchy's problem.- Sufficient conditions for the existence of energy estimates.- An analysis of conditions for the existence of energy estimates.- Cauchy's problem for dominantly correct differential operators.- References.