Film cooling has been selected for the thermal protection of the composite nozzle extension of the J-2X engine which is currently being developed for the second stage of NASA's next generation launch vehicle, the Ares I rocket. However, several challenges remain in order to achieve effective film cooling of the nozzle extension and to ensure its safe operation. The extreme complexity of the flow (three-dimensional wakes, lateral flows, vorticity, and flow separation) makes predicting film cooling performance difficult. There is also a dearth of useful supersonic film cooling data available for engineers to use in engine design and a lack of maturity of CFD tools to quantitatively match supersonic film cooling data. This dissertation advances the state of the art in film cooling by presenting semi-empirical analytical models which improve the basic physical understanding and prediction of the effects of pressure gradients, compressibility and density gradients on film cooling effectiveness. These models are shown to correlate most experimental data well and to resolve several conflicts in the open literature. The core-to-coolant stream velocity ratio, R, and the Kays acceleration parameter, K_P, are identified as the critical parameters needed to understand how pressure gradients influence film cooling performance. The convective Mach number, M_c, the total temperature ratio, Ω₀, and the Mach number of the high speed stream, M_HS, are shown to be important when explaining the effects of compressibility and density gradient on film cooling effectiveness. An advance in the simulation of film cooling flows is also presented through the development of a computationally inexpensive RANS methodology capable of correctly predicting film cooling performance under turbulent, subsonic conditions. The subsonic simulation results suggest that it in order to obtain accurate predictions using RANS it is essential to thoroughly characterize the turbulent states at the inlet of the coolant and core streams of the film cooling flow. The limitations of this approach are established using a Grid Convergence Index (GCI) Test and a demonstration of the extension of this RANS methodology to supersonic conditions is presented.