The article deals with multiperiodic differentiation operators with two, three and m+1 independent variables. The problems of bringing these operators to the canonical form are investigated. Conditions are found under which a multiperiodic operator turns into a canonical multiperiodic operator. The group property of the characteristics of a multiperiodic operator is determined. The relation defining zeros of the multiperiodic operator is found. Theorems on the reducibility of a multiperiodic differentiation operator to a linear operator with a narrow hyperbolic operator are proved. Following the idea of the works of Zh.A.Sartabanov, on the reduction of a quasilinear system to a canonical form, in this paper, a method has been developed for reducing a matrix differentiation operator with m+1 variables to a linear operator with a matrix differentiation operator with m variables, based on the transition along the characteristic of one of the independent variables.