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In this work, we consider the problem of Quality-Diversity (QD) optimization with multiple objectives. QD algorithms have been proposed to search for a large collection of both diverse and high-performing solutions instead of a single set of local optima. Thriving for diversity was shown to be useful in many industrial and robotics applications. On the other hand, most real-life problems exhibit several potentially antagonist objectives to be optimized. Hence being able to optimize for multiple objectives with an appropriate technique while thriving for diversity is important to many fields. Here, we propose an extension of the MAP-Elites algorithm in the multi-objective setting: Multi-Objective MAP-Elites (MOME). Namely, it combines the diversity inherited from the MAP-Elites grid algorithm with the strength of multi-objective optimizations by filling each cell with a Pareto Front. As such, it allows to extract diverse solutions in the descriptor space while exploring different compromises between objectives. We evaluate our method on several tasks, from standard optimization problems to robotics simulations. Our experimental evaluation shows the ability of MOME to provide diverse solutions while providing global performances similar to standard multi-objective algorithms.