This talk is a survey of the most important work of the well-known Georgian Mathematician Professor Giorgi Manjavidze ”Boundary value problems for analytic and generalized analytic functions”. There we present his original approach to subject. The work is devoted to boundary value problems of linear conjugation with shift.In these problems the boundary values of the desired functions are conjugated at points which are displaced to each other. The classical theory is extended to the case of generalized analytic functions and vectors. These problems have and will have important applications. The main attention is paid to the construction of the canonical matrices which are used in the construction of the general solutions of the considered problems. Explicit conditions of normal solvability and index formulas are obtained.