A complete Riemannian manifold is without conjugate points if every geodesic in its universal Riemannian covering is length-minimizing. Riemannian 2-tori without conjugate points are flat by a theorem of Eberhard Hopf from 1948. The present thesis contains rigidity results for complete Riemannian metrics without conjugate points on the plane and on the 2-cylinder. In these cases the area growth of the metric is a particularly natural condition and leads to optimal results. The thesis further contains rigidity results for the case of conformally flat cylinders of dimension three and greater.