Over the past decades, Unmanned Air Vehicles (UAVs) have achieved outstanding performance in military, commercial and civilian applications. UAVs are increasingly appearing in the form of swarms or formations to meet higher mission requirements. Communication plays an important role in UAV swarm control and coordination. The communication architecture defines how information is exchanged between UAVs or between UAVs and the central control center. Routing protocols help provide reliable end-to-end data transmission. Therefore, it is particularly important to design UAV swarm communication architectures and routing protocols with high performance and stability. This review article details four communication architectures including the advantages and disadvantages. Applicable scenarios are also discussed. In addition, a systematic overview and feasibility research of routing protocols are presented in this paper. To spur further research, the open research issues of UAV swarm communication architectures and routing protocols are also investigated.