This review surveys recent advances in the numerical modeling of solar prominences and coronal rain achieved with the fully open-source adaptive-grid, parallelized Adaptive Mesh Refinement Versatile Advection Code (MPI-AMRVAC). We examine how these models have contributed to our understanding of the formation and evolution of cool plasma structures in the solar corona. We first discuss prominence models that focus on prominence formation and their dynamic behaviour. We then turn to coronal rain, highlighting its connection to thermal instability and its role in the exchange of mass and energy between the corona and chromosphere. Particular attention is given to the growing efforts to connect simulations with observations through synthetic emission and spectral diagnostics.
