This thesis describes the design and development of a frequency doubled, continuous wave dye laser and its application to a study of the high Rydberg states of Rubidium. The laser uses the dye rhodamine 6G as the active medium and is optically pumped with an argon ion laser. Frequency doubling is by an ADA (ammonium dihydrogen arsenate) or ADP (ammonium dihydrogen phosphate) crystal located within the laser cavity. Continuous output powers in the ultra-violet in excess of 30 mW and tunable over the wavelength range 285-315 nm have been produced. The linewidth can be chosen to be 0.02 nm broadband system) or 0.002 nm (narrowband system) depending on the frequency selecting elements used. In order to keep insertion losses small the crystals have optical faces cut at Brewster's angle, and in order to increase generation efficiency the intracavity radiation field is focused into the crystal. Such an arrangement introduces the aberrations of coma and astigmatism which must be compensated by suitable cavity design. A variety of cavity and crystal configurations have been analysed for aberrations, and a novel arrangement for the simultaneous elimination of coma and astigmatism developed. Several practical frequency doubled dye lasers have been investigated. In particular the performances of ADA and ADP as the frequency doubling crystals are compared and contrasted. ADA has the advantage that it can be non-critically phase matched at these wavelengths and this results in a higher generation efficiency and a better UV beam quality than encountered with ADP. However, since it can only be temperature tuned, the tuning range (292-302 nm for temperature range 20-80°C) is more limited than that for ADP (285-315 nm) which can also be angled tuned. For both types of crystal, thermal phase mismatching is identified as the process limiting generation efficiency. Evidence is also presented that thermal focusing ultimately limits the UV output power by upsetting cavity stability. A computer model of intracavity frequency doubling has been developed. Thermal phase mismatching effects in the crystal as well as excited state absorption in the dye are included. This model is used to investigate the influence of cavity losses and crystal absorption on generation efficiency. Optimization of conversion efficiency by correct choice of crystal parameters is considered. Two systems have been developed to allow continuous scanning over an extended frequency range. One allows the broadband laser (0.02 nm) to be continuously tuned over 3 nm, the other allows the narrowband laser (0.002 nm) to be continuously tuned over 2 nm. The operation of a single frequency version of the laser and its stabilization on an external reference cavity is also described. The excitation of high Rydberg states in Rubidium using the frequency doubled laser is described. The states are detected by a space-charge limited ionization detector. The principal series of Rb up to a principal quantum number of n = 74 has been detected. A novel triode arrangement of electrodes in the space charge detector has enabled a small electric field to be applied to the rubidium vapour. The consequent Stark mixing of n2S, n2P and n2S states has allowed the n2S and n2D states to be excited from the 52S ground state. New term values of the n2S and n2D series are reported.