Diffusion models and large language models have emerged as leading-edge generative models, revolutionizing various aspects of human life. However, the practical implementations of these models have also exposed inherent risks, bringing to the forefront their evil sides and sparking concerns regarding their trustworthiness. Despite the wealth of literature on this subject, a comprehensive survey specifically delving into the intersection of large-scale generative models and their trustworthiness remains largely absent. To bridge this gap, this paper investigates both the long-standing and emerging threats associated with these models across four fundamental dimensions: 1) privacy, 2) security, 3) fairness, and 4) responsibility. Based on the investigation results, we develop an extensive map outlining the trustworthiness of large generative models. After that, we provide practical recommendations and potential research directions for future secure applications equipped with large generative models, ultimately promoting the trustworthiness of the models and benefiting the society as a whole.