A Riemann-Hilbert Approach to the Heun Equation

We describe the close connection between the linear system for the sixth Painlev'e equation and the general Heun equation, formulate the Riemann-Hilbert problem for the Heun functions and show how, in the case of reducible monodromy, the Riemann-Hilbert formalism can be used to construct explicit polynomial solutions of the Heun equation.