Abstract: We consider conformal immersions from a compact Riemann surface into the round 3-sphere with constant mean curvature (CMC surfaces). While the only CMC spheres are round spheres and CMC tori can be explicitly parametrized via integrable systems methods, only very little is known about higher genus CMC surfaces. In this talk I want first give a brief introduction to the integrable systems techniques for CMC tori due to Hitchin and Pinkall&Sterling. The key step is to translate the constant mean curvature condition into the flatness condition of a family of SL(2,C)-connections over the surface (which is independent of the genus of the surface). Then the abelian fundamental group of the torus allows to associate to the conformal immersion certain algebraic geometric objects - the spectral data - from which the conformal immersion can be reconstructed. Under the assumption of certain discrete symmetries, the family of flat connections can be reduced to a family of Fuchsian systems over the 4-punctured sphere, i.e., connections on a trivial C^2 bundle over the 2-sphere with simple poles at 4 points. The advantage of this picture is that the genus of the surface is encoded in the local monodromies of the residues of the Fuchsian system. Deforming these mondromies induces a flow from given spectral data of CMC tori to the spectral data of higher genus CMC surfaces through branched CMC immersions. I will give a proof for the short time existence of the flow at homogenous tori. This construction generalizes the flow of Plateau solutions from Clifford to Lawson's minimal surface of genus 2. Further, computer experiments indicate a correspondence between the moduli space of CMC tori and the moduli space of (symmetric) CMC surfaces of genus 2 via the flow. This is joint work with S. Heller, N. Schmitt.