The detection of irregularly spaced pulses of non-negligible width is a fascinating yet under-explored topic in signal processing. It sits adjacent to other core topics such as radar and symbol detection yet has its own distinctive challenges. Even modern techniques such as compressed sensing perform worse than may be expected on pulse processing problems. Real-world applications include nuclear spectroscopy, flow cytometry, seismic signal processing and neural spike sorting, and these in turn have applications to environmental radiation monitoring, surveying, diagnostic medicine, industrial imaging, biomedical imaging, top-down proteomics, and security screening, to name just a few. This overview paper endeavours to position the pulse processing problem in the context of signal processing. It also describes some current challenges in the field.