Multiplex networks describe a large number of complex social, biological and transportation networks where a set of nodes is connected by links of different nature and connotation. Here we uncover the rich community structure of multiplex networks by associating a community to each multilink where the multilinks characterize the connections existing between any two nodes of the multiplex network. Our community detection method reveals the rich interplay between the mesoscale structure of the multiplex networks and their multiplexity. For instance some nodes can belong to many layers and few communities while others can belong to few layers but many communities. Moreover the multilink communities can be formed by a different number of relevant layers. These results point out that mesoscopically there can be large differences in the compressibility of multiplex networks.