The rapid advancement of Generative Artificial Intelligence (GenAI) has significantly impacted game design, offering novel ways to enhance player experiences. GenAI is reshaping game design by introducing new opportunities for human-AI collaboration. This paper presents a review on how GenAI tools can be leveraged to create more meaningful, highly dynamic and personalized interactions within games. We begin by exploring the evolution of GenAI and its current applications in the gaming industry, highlighting its role in facilitating creative and immersive environments. A main area of investigation is the use of GenAI to enhance player experience, through procedural content generation, and adaptive gameplay mechanics. We examine how GenAI enables players to co-create within game worlds, enhancing their engagement. We delve into the customization of player experiences, analyzing how GenAI-driven systems can adapt to individual preferences and emotional responses to provide a tailored gaming experience. The paper also addresses the challenges and ethical concerns inherent in deploying GenAI in games, such as ensuring privacy, avoiding emotional manipulation, and providing appropriate content. By synthesizing findings from recent literature, this document offers insights about the integration of GenAI in game design to create enriched and personalized gaming experiences.