In this paper, we study the Dirac equation by Hörmander's \$L^2\$-method. For Dirac bundles over 2-dimensional Riemannian manifolds, in compact case we give a sufficient condition for the solvability of the Dirac equation in terms of a curvature integral; in noncompact case, we prove the Dirac equation is always solvable in weighted \$L^2\$ space. On compact Riemannian manifolds, we give a new proof of Bär's theorem comparing the first eigenvalue of the Dirac operator with that of the Yamabe type operator. On Riemannian manifolds with cylindrical ends, we obtain solvability in \$L^2\$ space with suitable exponential weights allowing mild negativity of the curvature. We also improve the above results when the Dirac bundle has a \$\mathbb{Z}_{2}\$ grading. Potential applications of our results are discussed. This is a joint work with Qingchun Ji.