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As high-performance computing (HPC) platforms continue to scale up, communication costs have become a critical bottleneck affecting overall application performance. An effective strategy to overcome this limitation is to overlap communication with computation. The Message Passing Interface (MPI), as the de facto standard for communication in HPC, provides non-blocking communication primitives that make such overlapping feasible. By enabling asynchronous communication, non-blocking operations reduce idle time of cores caused by data transfer delays, thereby improving resource utilization. Overlapping communication with computation is particularly important for enhancing the performance of large-scale scientific applications, such as numerical simulations, climate modeling, and other data-intensive tasks. However, achieving efficient overlapping is non-trivial and depends not only on advances in hardware technologies such as Remote Direct Memory Access (RDMA), but also on well-designed and optimized MPI implementations. This paper presents a comprehensive survey on the principles of MPI non-blocking communication, the core techniques for achieving computation–communication overlap, and some representative applications in scientific computing. Alongside the survey, we include a preliminary experimental study evaluating the effectiveness of asynchronous progress mechanism on modern HPC platforms to support the development of parallel programs for HPC researchers and practitioners.