With recent advances in artificial intelligence (AI) and robotics, unmanned vehicle swarms have received great attention from both academia and industry due to their potential to provide services that are difficult and dangerous to perform by humans. However, learning and coordinating movements and actions for a large number of unmanned vehicles in complex and dynamic environments introduce significant challenges to conventional AI methods. Generative AI (GAI), with its capabilities in complex data feature extraction, transformation, and enhancement, offers great potential in solving these challenges of unmanned vehicle swarms. For that, this paper aims to provide a comprehensive survey on applications, challenges, and opportunities of GAI in unmanned vehicle swarms. Specifically, we first present an overview of unmanned vehicles and unmanned vehicle swarms as well as their use cases and existing issues. Then, an in-depth background of various GAI techniques together with their capabilities in enhancing unmanned vehicle swarms are provided. After that, we present a comprehensive review on the applications and challenges of GAI in unmanned vehicle swarms with various insights and discussions. Finally, we highlight open issues of GAI in unmanned vehicle swarms and discuss potential research directions.