  We prove upper and lower bounds for the eigenvalues of the Dirac operator and
the Laplace operator on 2-dimensional tori. In particluar we give a lower bound
for the first eigenvalue of the Dirac operator for non-trivial spin structures.
It is the only explicit estimate for eigenvalues of the Dirac operator known so
far that uses information about the spin structure.
  As a corollary we obtain lower bounds for the Willmore functional of a torus
embedded into S_3.
  In the final section we compare Dirac spectra for two different spin
structures on an arbitrary Riemannian spin manifold.
