Collaboration between humans and robots (HRC) is advancing rapidly due to the intersection of robotics and generative artificial intelligence (GenAI). The current paper includes a systematic review of 103 studies on the role of generative models, including Generative Adversarial Networks (GANs), Variational Autoencoders (VAEs), diffusion models, and Large Language Models (LLMs) in improving the safety, trust, and adaptability of collaborative robotics using a PRISMA-based systematic approach. The review recognizes four major themed areas of GenAI-based safety frameworks—namely, data-driven simulation to synthesize hazards, predictive reasoning to forecast human motion, adaptive control to reduce risks dynamically, and trust-aware cognition to explain human–robot interaction. Findings indicate that generative models transform robotic safety from a reactive mechanism to proactive, contextual and interpretable systems. Nevertheless, real-time performance, interpretability, standard benchmarking, and ethical assurance are still some of the challenges to be overcome. The paper proposes a taxonomy linking generative modeling layers and physical, cognitive and ethical aspects of HRC safety, and gives a roadmap of certifiable hybrid systems with generative foresight and deterministic control. This synthesis provides a foundation for developing transparent, adaptive, and trustworthy collaborative robotic systems.