The integration of edge computing with Internet of Things (IoT) technology revolutionizes smart city services by enabling real-time data processing, decision-making, and action close to data sources. This paradigm shift addresses the limitations of traditional cloud-based models, which suffer from high latency, security risks, and limited bandwidth, especially in densely populated urban areas. Real-time edge computing offers a distributed approach to IoT data management by leveraging localized computing power at the network's edge, reducing the need for data to travel to centralized cloud systems. Edge computing facilitates efficient, responsive services in smart cities where applications like traffic management, environmental monitoring, public safety, and energy optimization demand immediate responses. Processing data locally enables quicker response times, enhances data privacy, and minimizes network congestion. This paper examines the architecture and technologies enabling edge computing in smart city applications and the unique challenges such as interoperability, scalability, and security. Case studies and implementations are explored to illustrate the transformative impact of real-time edge computing on urban infrastructure, contributing to more adaptive, resilient, and intelligent city ecosystems.