Relevance of the research is considered by the proof of correctness of the conjugation problem for linear hyperbolic and pseudoparabolic fourth-order equations with low figures. The main aim of the research is to prove the existence and uniqueness of solution for conjugation problem for hyperbolic and pseudoparabolic fourth-order equations when matching conditions are specified not on a characteristic line. The methods used in the study: Using Riman's function and integral equations methods and equivalent model the solution of the problem is reduced to the solution of the Fredgolm integral equations system of the second order where the solution is attained by the consequent approximation method. The results: The author has studied the solvability of the conjugation problem for hyperbolic and pseudoparabolic fourth-order equations with low variable coefficients. It was ascertained that when the equation order equals four, and matching conditions are specified not on the characteristic line, four gluing conditions should be specified for the problem correctness instead of usual two gluing conditions. The peculiarity of the problem is that the matching conditions are set not on the coordinate axis but on the bisector of the plane first quarter. In order to determine the unknown function trace and its derivatives of the second, third and fourth orders of changes in the line-type equations, as well as to obtain an explicit solution representation, Riemann functions were constructed for linear hyperbolic and pseudoparabolic fourth-order equations that are defined as solutions of the corresponding Goursat conjugate problems. The author studied some properties of the Riemann function and obtained the solution representations for the Cauchy problem for linear hyperbolic and pseudoparabolic fourth-order equations with variable coefficients. Solvability of the conjugation problem was obtained by its equivalent reduction to the solvability of the system of four linear Fredholm integral equations of the second kind. The theorems of existence and uniqueness of solutions of the conjugation problem for hyperbolic and pseudoparabolic fourth-order equations were proved.