Contextuality is a fundamental feature of quantum theory necessary for certain models of quantum computation and communication. Serious steps have therefore been taken towards a formal framework for contextuality as an operational resource. However, the main ingredient of a resource theory - a concrete, explicit form of free operations of contextuality– was still missing. Here we provide such a component by introducing noncontextual wirings: a class of contextuality-free operations with a clear operational interpretation and a friendly parametrization. We characterize them completely for general black-box measurement devices with arbitrarily many inputs and outputs. As applications, we show that the relative entropy of contextuality is a contextuality monotone and that maximally contextual boxes that serve as contextuality bits exist for a broad class of scenarios. Our results complete a unified resource-theoretic framework for contextuality and Bell nonlocality.