Unmeasured confounding is a threat to causal inference in observational studies. In recent years, the use of negative controls to mitigate unmeasured confounding has gained increasing recognition and popularity. Negative controls have a long-standing tradition in laboratory sciences and epidemiology to rule out non-causal explanations, although they have been used primarily for bias detection. Recently, Miao and colleagues have described sufficient conditions under which a pair of negative control exposure and outcome variables can be used to identify non-parametrically the average treatment effect (ATE) from observational data subject to uncontrolled confounding. We establish non-parametric identification of the ATE under weaker conditions in the case of categorical unmeasured confounding and negative control variables. We also provide a general semiparametric framework for obtaining inferences about the ATE while leveraging information about a possibly large number of measured covariates. In particular, we derive the semiparametric efficiency bound in the non-parametric model, and we propose multiply robust and locally efficient estimators when non-parametric estimation may not be feasible. We assess the finite sample performance of our methods in extensive simulation studies. Finally, we illustrate our methods with an application to the post-licensure surveillance of vaccine safety among children.