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Network embedding approaches are gaining momentum to analyse a large variety of networks. Indeed, these approaches have demonstrated their efficiency for tasks such as community detection, node classification, and link prediction. However, very few network embedding methods have been specifically designed to handle multiplex networks, i.e. networks composed of different layers sharing the same set of nodes but having different types of edges. Moreover, to our knowledge, existing approaches cannot embed multiple nodes from multiplex-heterogeneous networks, i.e. networks composed of several layers containing both different types of nodes and edges. In this study, we propose MultiVERSE, an extension of the VERSE method with Random Walks with Restart on Multiplex (RWR-M) and Multiplex-Heterogeneous (RWR-MH) networks. MultiVERSE is a fast and scalable method to learn node embeddings from multiplex and multiplex-heterogeneous networks. We evaluate MultiVERSE on several biological and social networks and demonstrate its efficiency. MultiVERSE indeed outperforms most of the other methods in the tasks of link prediction and network reconstruction for multiplex network embedding, and is also efficient in the task of link prediction for multiplex-heterogeneous network embedding. Finally, we apply MultiVERSE to study rare disease-gene associations using link prediction and clustering. MultiVERSE is freely available on github at https://github.com/Lpiol/MultiVERSE.