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Single trapped ions are ideal systems in which to test atomic physics at highprecision: they are effectively isolated atoms held at rest and largely freefrom perturbing interactions. This thesis describes several projects developedto study the structure of singly-ionized barium and more fundamental physics.  First, we describe a spin-dependent "electron-shelving" scheme that allows usto perform single ion electron spin resonance experiments in both the ground6S_{1/2} and metastable 5D_{3/2} states at precision levels of 10^{-5}. Weemploy this technique to measure the ratio of off-resonant light shifts (orac-Stark effect) in these states to a precision of 10^{-3} at two differentwavelengths. These results constitute a new high precision test of heavy-atomatomic theory. Such experimental tests in Ba+ are in high demand sinceknowledge of key dipole matrix elements is currently limited to about 5%. Ba+has recently been the subject of theoretical interest towards a test of atomicparity violation for which knowledge of dipole matrix elements is an importantprerequisite. We summarize this parity violation experimental concept anddescribe new ideas. (continued...)

Single barium ion spectroscopy: light shifts, hyperfine structure, and progress on an optical frequency standard and atomic parity violation