Cyclic spectral analysis is a signal-processing technique designed to deal with stochastic signals whose statistics vary periodically with time. Pulsar radio emission is a textbook example of this signal class, known as cyclostationary signals. In this paper, we discuss the application of cyclic spectral analysis methods to pulsar data and compare the results with the traditional filterbank approaches used for almost all pulsar observations to date. In contrast to standard methods, the cyclic spectrum preserves phase information of the radio signal. This feature allows us to determine the impulse response of the interstellar medium and the intrinsic, unscattered pulse profile directly from a single observation. We illustrate these new analysis techniques using real data from an observation of the millisecond pulsar B1937+21.