The success of deep neural networks has driven numerous research studies and applications from Euclidean to non-Euclidean data. However, there are increasing concerns about privacy leakage, as these networks rely on processing private data. Recently, a new type of privacy attack, the model inversion attacks (MIAs), aims to extract sensitive features of private data for training by abusing access to a well-trained model. The effectiveness of MIAs has been demonstrated in various domains, including images, texts, and graphs. These attacks highlight the vulnerability of neural networks and raise awareness about the risk of privacy leakage within the research community. Despite the significance, there is a lack of systematic studies that provide a comprehensive overview and deeper insights into MIAs across different domains. This survey aims to summarize up-to-date MIA methods in both attacks and defenses, highlighting their contributions and limitations, underlying modeling principles, optimization challenges, and future directions. We hope this survey bridges the gap in the literature and facilitates future research in this critical area. Besides, we are maintaining a repository to keep track of relevant research at https://github.com/AndrewZhou924/Awesome-model-inversion-attack.