This dissertation presents a detailed study of many aspects of the Horizontal Branch (HB) phase of stellar evolution. A classical technique of stellar structure analysis is summarized, and applied to Zero-Age Horizontal Branch (ZAHB) models. The chief conclusions from this work are firstly, that the total mass of the envelope sensitively affects the luminosity of the hydrogen-burning shell and the equilibrium of the helium-rich core. Secondly, the rapid progression of models across the Hertzsprung-Russell diagram with decreasing mass is the result of important changes in the hydrostatic structure of the stars. Thirdly, the luminosity-metallicity relationship of the Zero Age models results from the change in the core equilibrium luminosity with the CNO abundance of the shell region, together with the decrease in stellar mass at fixed effective temperature. The change in the mass-temperature relation with CNO is found to be the most important determining factor in the Horizontal Branch stellar distribution, and therefore is the most appropriate ‘first parameter’ for HB morphology. The evolution of the stars is then considered, and the analysis of the interior structures provides a reclassification of HB track morphology into three categories, depending on whether the model contains an outer convection zone or a radiative outer envelope, and on the luminosity of the hydrogen-burning shell. Lastly, the question of the formation of red-giant stars is considered; the general conclusions of this part of the study support the arguments presented by Yahil and van den Horn (1985). Next, the evolution of the convective core of HB stars is reviewed, together with a detailed account of the numerical techniques developed for modelling semi-convection. The problems associated with the late phase of HB evolution are also discussed. A brief review of the physical inputs and numerical methods used in the interior is presented, focussing on the calculation and implementation of the Equation of State. The calculations performed for this study are then presented in detail. The effects of oxygen enhancement on zero-age sequences are illustrated for a range in metallicity, and theoretical relations between luminosity and metallicity for the ZAHBs are demonstrated. The evolutionary tracks computed are illustrated and summarized in extensive tabulations in the Appendices. The final chapter reproduces previously published studies of globular clusters. The first of these investigates the globular cluster NGC104 (47 Tucanae). By fitting the theoretical models to recent CCD photometry of the cluster, it was found that its initial helium content must have been close to 24% by mass. In addition, the best fits show that models for [Fe/H] = -0.65 provide an excellent match to the horizontal branch, if (m - M )v ≈ 13.44, and thereby yield consistency over the entire color-magnitude diagram of the cluster. The second study presents an investigation of the horizontal branch of M15. Detailed matches of our theoretical sequences to the cluster observations indicate that high envelope helium abundances are incompatible with the observed morphology. It is found that there is a clear preference for values of 0.21 ≾ Y ≾ 0.25, independent of the value of [O/Fe]. The precision of the method is reduced by uncertainties in the observations and in the available synthetic temperature-bolometric-correction relations. The oxygen enhanced zero-age HB models are found to have a period-colour relationship which is almost identical to that of their scaled-solar counterparts, but they reduce significantly the predicted double-mode variable masses. Importantly, it is found that, for reasonable assumptions about the reddening to M15, there is no discrepancy between the predicted and observed periods for the RR Lyrae variables. However, the period shift between M3 and M15 can be explained by canonical models only if the helium abundance in both clusters is low (Yhb ~ 0.21), and the bulk of the RR Lyrae star population in M15 is at late stages of evolution. These conclusions are reconsidered in the light of the new calculations presented here. / Graduate