The application of radiocarbon dating is extended to include systems that are slowly exchanging carbon with the atmosphere. Simple formulae are derived that relate the true age and the exchange rate of carbon to the apparent radiocarbon age. A radiocarbon age determination does not give a unique true age and exchange rate but determines a locus of values bounded by a minimum age and a minimum exchange rate. It is found that for radiocarbon ages as large as 10,000 years it is necessary to correct for the anthropogenic radiocarbon produced in the atmosphere by nuclear weapons testing. A one-term exponential approximation, with an e-folding time of 14.43 years, is used to model this effect and is shown to be accurate to within 3% for exchange time constants of 100 years and greater. The approach developed here is not specific to radiocarbon and can be applied to other radioisotopes in open systems.