A Multilayer Network (MN) is a system consisting of several topological levels (i.e., layers) representing the interactions between the system's objects and the related interdependency. Therefore, it may be represented as a set of layers that can be assimilated to a set of networks of its own objects, by means inter-layer edges (or inter-edges) linking the nodes of different layers; for instance, a biological MN may allow modeling of inter and intra interactions among diseases, genes, and drugs, only using its own structure. The analysis of MNs may reveal hidden knowledge, as demonstrated by several algorithms for the analysis. Recently, there is a growing interest in comparing two MNs by revealing local regions of similarity, as a counterpart of Network Alignment algorithms (NA) for simple networks. However, classical algorithms for NA such as Local NA (LNA) cannot be applied on multilayer networks, since they are not able to deal with inter-layer edges. Therefore, there is the need for the introduction of novel algorithms. In this paper, we present MuLaN, an algorithm for the local alignment of multilayer networks. We first show as proof of concept the performances of MuLaN on a set of synthetic multilayer networks. Then, we used as a case study a real multilayer network in the biomedical domain. Our results show that MuLaN is able to build high-quality alignments and can extract knowledge about the aligned multilayer networks. MuLaN is available at https://github.com/pietrocinaglia/mulan.