The digital transformation of modern cities by integrating advanced information, communication, and computing technologies has marked the epoch of data-driven smart city applications for efficient and sustainable urban management. Despite their effectiveness, these applications often rely on massive amounts of high-dimensional and multi-domain data for monitoring and characterizing different urban sub-systems, presenting challenges in application areas that are limited by data quality and availability, as well as costly efforts for generating urban scenarios and design alternatives. As an emerging research area in deep learning, Generative Artificial Intelligence (GenAI) models have demonstrated their unique values in content generation. This paper aims to explore the innovative integration of GenAI techniques and urban digital twins to address challenges in the planning and management of built environments with focuses on various urban sub-systems, such as transportation, energy, water, and building and infrastructure. The survey starts with the introduction of cutting-edge generative AI models, such as the Generative Adversarial Networks (GAN), Variational Autoencoders (VAEs), Generative Pre-trained Transformer (GPT), followed by a scoping review of the existing urban science applications that leverage the intelligent and autonomous capability of these techniques to facilitate the research, operations, and management of critical urban subsystems, as well as the holistic planning and design of the built environment. Based on the review, we discuss potential opportunities and technical strategies that integrate GenAI models into the next-generation urban digital twins for more intelligent, scalable, and automated smart city development and management.