Generative modeling in machine learning aims to synthesize new data samples that are statistically similar to those observed during training. While conventional generative models such as GANs and diffusion models typically assume access to large and diverse datasets, many real-world applications (e.g. in medicine, satellite imaging, and artistic domains) operate under limited data availability and strict constraints. In this survey, we examine Generative Modeling under Data Constraint (GM-DC), w...