The contact of a deformable fluid surface with a deformable solid surface gives rise to a rich variety of phenomena, including topological changes. In this talk, I will present a computational model which is capable to shed some light on these processes. The numerical method is based on a combination of the phase-field model for two-phase flow and an evolving finite-element method for the elastic structure. We demonstrate the flexibility and accuracy of this novel method and connect the numerical results to recent findings in cell biology, where the interaction of droplets with biological membranes produces intriguing phenomena.