The paper provides a comprehensive review of the recent advancements and methodologies in Human-Robot Collaboration (HRC) applied to the manufacturing assembly process. In modern manufacturing, the assembly process involves intricate and time-consuming operations, often necessitating flexible manual interventions. However, the cost and stability issues associated with manual labor highlight the need for collaborative solutions integrating humans and robots. HRC, as a viable solution, involves the joint effort of humans and robots in manufacturing tasks, presenting advantages in terms of precision, reproducibility, and cycle time. This review categorizes and discusses methodologies such as task allocation, reinforcement learning, and Cyber-Physical Systems (CPS)-based planning approaches that facilitate HRC in the assembly process. It also explores experiments and future trends to address challenges and enhance efficiency in manufacturing assembly through intelligent collaboration between humans and robots. The objective of this research is to provide insights and directions for further research in HRC to optimize manufacturing processes. By analyzing the existing state-of-the-art and presenting future prospects, this paper aims to guide researchers and practitioners toward more effective implementations of HRC in manufacturing assembly, ultimately leading to improved operational efficiency and productivity.