The solutions for all long operators of the N=4 SYM theory at one loop
and of its dual, the classical superstring theory
on AdS5xS5 space, will be presented in terms of their algebraic curves
encoding all conservation laws. The SYM theory is treated by the Bethe
ansatz equations in the thermodynamical limit, whereas the
classical superstring is solved by the finite gap method, using the
integrability. In a special (Frolov-Tseytlin) limit both curves
coincide, demonstrating the full AdS/CFT correspondence among
states/operators, including fermions.