Urban transportation networks are increasingly strained by growing populations and expanding urbanization. Intelligent transportation systems (ITS) have emerged to address these challenges, leveraging advanced technologies to enhance urban mobility's efficiency, safety, and sustainability. Public transportation, particularly bus systems, plays a critical role in urban mobility, offering a sustainable alternative to private vehicles and reducing traffic congestion. ITS addresses numerous challenges associated with traditional transit networks, such as traffic congestion, pollution, improper planning of routes and schedules, economic barriers, etc. In this article, we studied the evolution of approaches for optimizing schedules and routes with reference to network development schedules and route optimization. Some key solution approaches to tackle the challenges in the mentioned urban bus transportation system areas, like the exact, metaheuristics, agent-based, and exact approaches, are discussed with their respective limitations. The advancements in these approaches include the integration of real-time data analytics for dynamic scheduling and routing, the deployment of Internet of Things (IoT) sensors for vehicle and infrastructure monitoring, and the application of artificial intelligence (AI) and machine learning (ML) algorithms for predictive maintenance and demand forecasting.