Optical absorption spectroscopy enables atoms, molecules and media to be characterised in the chemical, physical, biological and earth sciences. It yields a dependable and highly sensitive means of qualitative and quantitative analysis in many contexts: laboratory, environmental, industrial, clinical and beyond. 'Cavity-based' variants of absorption spectroscopy employ an optical cavity (usually resonant with one or more of the wavelengths of radiation used to observe absorption) to enhance the sensitivity (and, necessarily, the signal-to-noise ratio) by orders of magnitude relative to absorption spectroscopy performed without such a cavity. We offer general technical insights into various forms of cavity-based absorption spectroscopy, including: high-performance benchmarks such as NICE-OHMS; cavity-ringdown spectroscopy cavity-enhanced absorption spectroscopy intracavity laser absorption spectroscopy. We outline selected applications and indicate areas of current scientific interest and opportunity in this rapidly expanding field.