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A definition of a recurrent code is given in a framework which renders it amenable to mathematical analysis. Recurrent codes for both independent and burst errors are considered, and a necessary and sufficient condition for either type of error correction is established. For burst-error-correcting codes, the problem treated is (for a fixed burst length and redundancy) the minimization of the error-free distance ("guard space") required between bursts. A lower bound is obtained on the guard space, and in certain cases, codes which realize this bound are given. A general code which is close to the lower bound in many cases is also given. For independent errors, a code which will correct any error, provided that no consecutive "