  In this paper we will prove new extrinsic upper bounds for the eigenvalues of
the Dirac operator on an isometrically immersed surface $M^2 \hookrightarrow
{\Bbb R}^3$ as well as intrinsic bounds for 2-dimensional compact manifolds of
genus zero and genus one. Moreover, we compare the different estimates of the
eigenvalue of the Dirac operator for special families of metrics.
