Boundary-value problems with a shift for analytic functions and singular functional equations

We consider methods of studying the fundamental boundary-value problems with a shift, in the plane and on a Riemann surface, and of studying singular integro-functional equations with a shift.
In § § 1–3 we present an application of the method of conformal pasting to problems with a shift on a Riemann surface.
In § 4 we present the classical method of integral equations, applied to one of the problems (of the Carleman type).
In § § 5 and 6 we study singular integral equations with a shift satisfying the Carleman condition, and the corresponding general boundary-value problems. The fundamental method of reducing the problem to a system of singular equations with a Cauchy kernel and then applying the theorem on the stability of the index, allows us to obtain conditions for the problem to be noetherian and to calculate its index. In § 6 we introduce the concept of the stability of a problem with a Carleman shift, which is analogous to the concept of the stability of the partial indices of the Riemann problem; we establish the sufficiency of a stability criterion for the problem of Markushevich.
At the end of § 6 and in § 7 we survey papers on the subject that have not been discussed in the main part of the article.