We present AESim, a data-driven Aircraft Engine Simulator developed using transformer-based conditional generative adversarial networks. AESim generates samples of aircraft engine sensor measurements over full flights, conditioned on a given flight mission profile representing the flight conditions. It constitutes an essential tool in aircraft engine digital twins, capable of simulating their performance for different flight missions. It allows for comparison of the behavior of different engines under the same operational conditions, simulation of various scenarios for a given engine, facilitating applications like engine behavior analysis, performance limit identification, and optimization of maintenance schedules within a global Prognostics and Health Management (PHM) strategy. It also allows the imputation of missing flight data and addresses confidentiality concerns by generating synthetic flight datasets that can be shared for public research purposes or data challenges.