This volume develops the method of Newton’s polyhedron for solving some problems in the theory of partial differential equations. The content is divided into two parts. Chapters 1-4 consider Newton’s polygon makes it possible not only to consider general constructions in the two-dimensional case, but also leads to some natural multidmensional applications. Attention is mainly focused on a special class of hypoelliptic operators defined using Newton’s polyhedron, energy estimates in Cauchy’s problem relating to Newton’s polyhedron, energy estimates in Cauchy’s problem relating to Newton’s polyhedron, and generalized operators of principal type. Priority is given to the presentation of an algebraic technique which can be applied to many other problems as well. For researchers and graduate students whose work involves the theory of differential and pseudodifferential equations.