Intelligent Transportation Systems (ITS) operate within a highly intricate and dynamic environment characterized by complex spatial and temporal dynamics at various scales, further compounded by fluctuating conditions influenced by external factors such as social events, holidays, and weather conditions. Navigating the intricacies of modeling the intricate interaction among these elements, creating universal representations, and employing them to address transportation issues presents a significant endeavor. Yet, these intricacies comprise just one facet of the multifaceted trials confronting contemporary ITS. This paper offers an all-encompassing survey exploring Deep learning (DL) utilization in ITS, primarily focusing on practitioners' methodologies to address these multifaceted challenges. The emphasis lies on the architectural and problem-specific factors that guide the formulation of innovative solutions. In addition to shedding light on the state-of-the-art DL algorithms, we also explore potential applications of DL and large language models (LLMs) in ITS, including traffic flow prediction, vehicle detection and classification, road condition monitoring, traffic sign recognition, and autonomous vehicles. Besides, we identify several future challenges and research directions that can push the boundaries of ITS, including the critical aspects of explainability, transfer learning, hybrid models, privacy and security, and ultra-reliable low-latency communication. Our aim for this survey is to bridge the gap between the burgeoning DL and transportation communities. By doing so, we aim to facilitate a deeper comprehension of the challenges and possibilities within this field. We hope that this effort will inspire further exploration of fresh perspectives and issues, which, in turn, will play a pivotal role in shaping the future of transportation systems.