The authors examine the space of Riemann surfaces of signature (1,1) with metric of curvature -1 and geodesic boundary. They solve explicitly the moduli problem in this case and show furthermore that two surfaces of this type having the same length spectrum (this referring to smooth closed geodesics including the boundary) are isometric. They announce the same type of result for genus two surfaces without boundary. The problem whether the analogous assertion holds for the spectrum of the Laplacian with respect to the Neumann or Dirichlet conditions is open.