Given a valued field (K,v) with valuation ring O and an algebraic group G over K, a model of G is a group scheme \mathcal{G} over O for which \mathcal{G} \times_O K=G. We exhibit the existence of such models for G an elliptic curve and K an algebraically closed field, and on the way we classify the different group schemes one can get. The proof uses a combination of tools from model theory and algebraic geometry. The aim of this talk is to state the theorem and to introduce some of the model-theoretic notions required for the proof. No prior knowledge in model theory is required, though a basic course in logic will help.