Heterogeneous networks (HetNets) merge different types of networks into an integrated network system, which has become a hot research area in recent years towards nextgeneration communication networks. HetNets aim to effectively exploit network resources and provide seamless connectivity for heterogeneous objects. Unlike other networks, HetNets hold such characteristics as heterogeneity, openness, distribution, multidomain involvement, thus are susceptible to various security threats and attacks. Traditional security approaches are not sufficiently effective in defending against them. With extensive study and practice, researchers found that trust models offer effective measures to enhance the security and reliability of a network system. However, there still lacks a comprehensive survey on the recent advances of trust models in HetNets. In this paper, we fill this gap. We first retrospect the history of HetNets research and introduce important concepts related to trust. Then, we propose a set of criteria that a sound trust model should satisfy, which can also serve as a measure to evaluate the quality of a trust model, i.e., Quality of Trust (QoT). We provide taxonomies of trust models and their applications, and continue with a thorough review on trust models in HetNets. Based on the review, a list of open issues is highlighted, and corresponding future research directions are suggested to advance the research on trustworthy HetNets.