This memoir develops the spectral theory of the Lax operators of nonlinear Schrödinger-like partial differential equations with periodic boundary conditions. Their spectral curves, i.e., the common spectrum with the periodic shifts, are generically Riemann surfaces of infinite genus. The points corresponding to infinite energy are added. The resulting spaces are no longer Riemann surfaces in the usual sense, but they are quite similar to compact Riemann surfaces. In fact, some of the basic tools of the theory of compact Riemann surfaces are generalized to these spectral curves and illuminate the structure of complete integrability:

• The eigen bundles define holomorphic line bundles on the spectral curves, which completely determine the potentials.
• These line bundles may be described by divisors of the same degree as the genus, and these divisors give rise to Darboux coordinates.
• With the help of a Riemann-Roch Theorem, the isospectral sets (the sets of all potentials corresponding to the same spectral curve) may be identified with open dense subsets of the Jacobian varieties.
• The real parts of the isospectral sets are infinite dimensional tori, and the group action solves the corresponding nonlinear partial differential equations.
• Deformations of the spectral curves are in one to one correspondence with holomorphic forms.
• Serre Duality reproduces the symplectic form.