In recent years, with the advancement of science and technology, complex networks have grown exponentially in scale and complexity. Research in multiple fields in complex networks has gradually become a hotspot in multidisciplinary research. Among them, the community structure is an important field widely studied in complex networks, and it is also one of the important characteristics of complex networks. The community in the network is a collection of densely connected vertices, which are tightly connected within the community and sparsely connected with nodes outside the community. Community discovery plays an important role in understanding the functions of complex networks. At present, the research on homogeneous networks based on a single relationship type is relatively mature. However, most networks in the real world are heterogeneous networks with multiple relationships and multiple types. At the same time, as the scale of data continues to expand and demand changes, the research on heterogeneous networks has gradually become a hot spot in this field. This article summarizes the definitions and evaluation indicators of problems found in communities under heterogeneous networks. Then, according to the different methods, the existing community discovery algorithms under the heterogeneous network are summarized and sorted out, and then various methods are explained in detail by category. Secondly, related applications discovered by the community are introduced, including social network influence, link prediction, etc. And organize the commonly used data sets for researchers'' reference. Finally, the development trend of community discovery is summarized and prospected.