This document presents a study of the numerical simulation of non-equilibrium plasma discharges in air mixtures in the atmospheric pressure regime. Such plasma is formed by applying a very high electric field over a very short time duration (nano-microsecond) which preferentially heats the electrons to very high temperatures (10 electron Volts or more) while preventing thermalization of the gas. Preferentially heating the electrons to very high temperatures allows the discharge to efficiently and rapidly ionize and dissociate the gas mixture without losing too much energy to thermalization or vibrational excitation. Consequently, two useful characteristics of these discharges are low gas temperatures and rapid electron chemistry. This study focuses on two applications of interest: ignition of fuel-air mixtures and plasma enhanced medicine. For ignition, there are two situations that arise where it is difficult for traditional spark ignition systems to operate. The first is at the supersonic flow regime where the residence time of the flow in the engine is low. The second is high pressure ignition of lean fuel-air mixtures. For plasma medicine and surface treatment, non-equilibrium plasma is an effective means of delivering reactive radical species to the surface while limiting damage due to thermal heating. The problems of interest are characterized by the formation of weakly ionized plasma in the presence of flow fields such as supersonic boundary layers or low speed jets. To simulate the coupled plasma-fluid flow physics of these discharges, two numerical tools are utilized. The first is a two-temperature, multiple species, self-consistent plasma solver with finite rate chemistry which is used to simulate the plasma as it forms in a neutral background gas. The second tool is a multiple-species compressible flow solver which calculates the flow field properties of the background gas mixture.