Abstract: In this article, the authors discuss the Riemann boundary value problems with given principal part. First, authors consider a special case and give a classification of the solution class R_n by the way. And then, they consider the general case. The solvable conditions for this problem and its solutions is obtained when it is solvable.

Key words: Riemann boundary value problems, principal part, Riemann-Hilbert analysis for orthogonal polynomials