The denial of effective communications by enemy forces during hostile military operations has been a primary concern for military commanders since the inception of radio communications on the battlefield before World War II. Since then, the electromagnetic environment has been in a constant state of evolution toward more sophisticated jam-resistant and convert forms of modulation. For example, exotic modulation techniques employing spread spectrum (SS) signaling are routinely used by our adversaries to provide their communication links an advantage over US and Allied jammers. These same spread spectrum modulation techniques are being refined to provide convert, low probability-of-intercept (LPI) features to the unintended interceptor. The thrust of this paper focuses on developments in the theory and algorithms for detection, characterization, and exploitation of advanced waveforms using new mathematical signal processing tools. Specifically, quadratic time-frequency signal representations, wavelet transforms, and cyclostationary signal processing are introduced. This overview demonstrates the importance of these advanced techniques in a clear and concise manner. Applications and future research activities are described.