Spectral analysis and ARMA models have been the most common weapons of choice for the detection of cycles in political time series. Controversies about cycles, however, tend to revolve around an issue that both techniques are badly equipped to address: the possibility of irregular cycles without fixed periodicity throughout the entire time series. This has led to two main consequences. On the one hand, proponents of cyclical theories have often dismissed established statistical techniques. On the other hand, proponents of established techniques have dismissed the possibility of cycles without fixed periodicity. Wavelets allow the detection of transient and coexisting cycles and structural breaks in periodicity. In this article, we present the tools of wavelet analysis and apply them to the study of two lingering puzzles in the political science literature: the existence of cycles in election returns in the United States and in the severity of major power wars.