A new photoabsorption technique for measuring the spectra of trace amounts of species with both high sensitivity and high spectral resolution is described. The method is based on measurement of the time rate of decay of a pulse of light trapped in a high reflectance optical cavity. In this review paper, we present an account of the basic theory, development, and applications of this method to samples in the gas, liquid and solid phases. In addition to a general review, experimental data are presented that demonstrate the simplicity, sensitivity, and generality of this powerful spectroscopic tool.