This paper proposes the first model-based clustering algorithm for multivariate functional data. After introducing multivariate functional principal components analysis (MFPCA), a parametric mixture model, {based on the assumption of normality of the principal components}, is defined and estimated by an EM-like algorithm. The main advantage of the proposed model is its ability to take into account the dependence among curves. Results on simulated and real datasets show the efficiency of the proposed method.