We propose an augmented Lagrangian method for the solution of constrained optimization problems in Banach spaces. The framework we consider is very general and encompasses a host of standard problems such as nonlinear programming, semidefinite programming, and optimal control. We analyze several convergence-related aspects of the method, including global convergence, the attainment of feasibility, and local convergence. In particular, a result is presented which guarantees local convergence of the method under the sole assumption that a solution of the problem satisfies the second-order sufficient condition. Finally, we give detailed numerical results for optimal control problems and mathematical programs with complementarity constraints.