This survey paper explores the transformative influence of frontier AI, foundation models, and Large Language Models (LLMs) in the realm of Intelligent Transportation Systems (ITS), emphasizing their integral role in advancing transportation intelligence, optimizing traffic management, and contributing to the realization of smart cities. Frontier AI refers to the forefront of AI technology, encompassing the latest advancements, innovations, and experimental techniques in the field, especially AI foundation models and LLMs. Foundation models, like GPT-4, are large, general-purpose AI models that provide a base for a wide range of applications. They are characterized by their versatility and scalability. LLMs are obtained from finetuning foundation models with a specific focus on processing and generating natural language. They excel in tasks like language understanding, text generation, translation, and summarization. By leveraging vast textual data, including traffic reports and social media interactions, LLMs extract critical insights, fostering the evolution of ITS. The survey navigates the dynamic synergy between LLMs and ITS, delving into applications in traffic management, integration into autonomous vehicles, and their role in shaping smart cities. It provides insights into ongoing research, innovations, and emerging trends, aiming to inspire collaboration at the intersection of language, intelligence, and mobility for safer, more efficient, and sustainable transportation systems. The paper further surveys interactions between LLMs and various aspects of ITS, exploring roles in traffic management, facilitating autonomous vehicles, and contributing to smart city development, while addressing challenges brought by frontier AI and foundation models. This paper offers valuable inspiration for future research and innovation in the transformative domain of intelligent transportation.