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Offline reinforcement learning methods hold the promise of learning policies from pre-collected datasets without the need to query the environment for new transitions. This setting is particularly well-suited for continuous control robotic applications for which online data collection based on trial-and-error is costly and potentially unsafe. In practice, offline datasets are often heterogeneous, i.e., collected in a variety of scenarios, such as data from several human demonstrators or from policies that act with different purposes. Unfortunately, such datasets can exacerbate the distribution shift between the behavior policy underlying the data and the optimal policy to be learned, leading to poor performance. To address this challenge, we propose to leverage latent-variable policies that can represent a broader class of policy distributions, leading to better adherence to the training data distribution while maximizing reward via a policy over the latent variable. As we empirically show on a range of simulated locomotion, navigation, and manipulation tasks, our method referred to as latent-variable advantage-weighted policy optimization (LAPO), improves the average performance of the next best-performing offline reinforcement learning methods by 49% on heterogeneous datasets, and by 8% on datasets with narrow and biased distributions.