With the significant advancement in emerging processor, memory, and networking technologies, exascale systems will become available in the next few years (2020–2022). As the exascale systems begin to be deployed and used, there will be a continuous demand to run next-generation applications with finer granularity, finer time-steps, and increased data sizes. Based on historical trends, next-generation applications will require postexascale systems during 2025–2035. In this study, we focus on the networking and communication challenges for post-exascale systems. Firstly, we present an envisioned architecture for post-exascale systems. Secondly, the challenges are summarized from different perspectives: heterogeneous networking technologies, high-performance communication and synchronization protocols, integrated support with accelerators and field-programmable gate arrays, fault-tolerance and quality-of-service support, energy-aware communication schemes and protocols, softwaredefined networking, and scalable communication protocols with heterogeneous memory and storage. Thirdly, we present the challenges in designing efficient programming model support for high-performance computing, big data, and deep learning on these systems. Finally, we emphasize the critical need for co-designing runtime with upper layers on these systems to achieve the maximum performance and scalability.