  We investigate the monodromy of the Lax connection for classical IIB
superstrings on AdS_5xS^5. For any solution of the equations of motion we
derive a spectral curve of degree 4+4. The curve consists purely of conserved
quantities, all gauge degrees of freedom have been eliminated in this form. The
most relevant quantities of the solution, such as its energy, can be expressed
through certain holomorphic integrals on the curve. This allows for a
classification of finite gap solutions analogous to the general solution of
strings in flat space. The role of fermions in the context of the algebraic
curve is clarified. Finally, we derive a set of integral equations which
reformulates the algebraic curve as a Riemann-Hilbert problem. They agree with
the planar, one-loop N=4 supersymmetric gauge theory proving the complete
agreement of spectra in this approximation.
