Non-linear MHD simulations play an essential role in active research and understanding of tokamak plasmas for fusion energy. The development of MHD codes like JOREK is a key aspect of this research effort. In this paper, we present a fully-working version of the full-MHD model in JOREK, a significant advancement from the reduced-MHD model used for previous studies. The model is presented in details, and tests are performed using both linear and non-linear simulations. It is shown that this new model is numerically and physically reliable, without the necessity of numerical stabilisation methods. Typical tokamak instabilities are addressed, including disruption and ELM physics, most relevant to current open issues concerning future tokamaks like ITER and DEMO.