This paper reports a numerical investigation of droplet evaporation in a supercritical environment. A comprehensive physical-numerical model is developed to simulate the transcritical and supercritical droplet vaporization phenomena. The model is based on the time-dependent conservation equations for both liquid and gas phases, pressure-dependent variable thermophysical properties, and a detailed treatment of liquid-vapor phase equilibrium at the droplet surface. The numerical solution of the two-phase equations employs an arbitrary Eulerian-Lagrangian, explicit-implicit method with a dynamically adaptive mesh. The first part of the study examines the capability of different equations of state (EOS) for predicting the phase equilibrium and transcritical droplet vaporization behavior. Predictions using the Redlich-Kwong (RK) EOS are shown to be markedly different from those using the Peng-Robinson (PR) and Soave-Redlich-Kwong (SRK) EOS. Results for the phase-equilibrium of a n-heptane-nitrogen system indicate that compared to PR- and SRK-EOS, the RK-EOS predicts higher fuel vapor concentration, higher liquid-phase solubility of nitrogen, lower critical-mixing-state temperature, and lower enthalpy of vaporization. As a consequence, it significantly overpredicts droplet vaporization rates and, thus, underpredicts droplet lifetimes compared to those predicted by PR- and SRK-EOS, as well as compared to experimental data. Furthermore, using RK-EOS, attainment of the critical mixing state at the droplet surface is predicted earlier in droplet lifetime compared with that using the other two EOS. In contrast, predictions using the PR-EOS show excellent agreement with experimental data over a wide range of ambient conditions. The PR-EOS is then used for a detailed investigation of the transcritical droplet vaporization phenomena. Results indicate that at low to moderate ambient temperatures, the droplet lifetime first increases and then decreases as the ambient pressure is increased, while at high ambient temperatures, the droplet lifetime decreases monotonically with pressure. This behavior is in accord with the published experimental results. The numerical model is also used to obtain the minimum pressure required for the attainment of critical mixing state at the droplet surface for a given ambient temperature.