We have used stored ion methods to improve resolution and sensitivity in optical spectroscopy. Single atomic ions have been confined by electric and magnetic fields, cooled by laser radiation pressure to temperatures on the order of 1 mK, and probed spectroscopically with narrowband lasers. The absorption resonance of a single Hg+ ion has been observed by a decrease in the transmitted light intensity. An ultraviolet transition in Hg+ has been observed with a linewidth of only 30 kHz. Quantum jumps to and from metastable levels of Hg+ have been observed and used to determine radiative decay rates and to infer the existence of photon antibunching. Quantum jumps have also been observed in single Mg+ ions.