Abstract. The Adaptive Multiple Importance Sampling algorithm is aimed at an optimal recycling of past simulations in an iterated importance sampling (IS) scheme. The difference with earlier adaptive IS implementations like Population Monte Carlo is that the importance weights of all simulated values, past as well as present, are recomputed at each iteration, following the technique of the deterministic multiple mixture estimator of Owen & Zhou (J. Amer. Statist. Assoc., 95, 2000, 135). Although the convergence properties of the algorithm cannot be investigated, we demonstrate through a challenging banana shape target distribution and a population genetics example that the improvement brought by this technique is substantial.