With the significant advancement in emerging processor,memory,and networking technologies,exascale systems will become available iu 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 applicatious 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 programmiug model support for high-performance computing,big data,and deep leaing 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.