Lazy wave risers (LWRs) are designed with equidistant buoyancy blocks attached in the lower half of the riser, allowing the riser to take on an arch shape under the buoyancy forces provided by buoyancy blocks. This arch configuration can provide flexibility to the LWR arrangement and effectively isolate the dynamic responses between the offshore floating structure and the riser’s touchdown zone (TDZ). Its design and application aim to address the issues of dynamic response and fatigue damage that traditional steel catenary risers (SCRs) face in deep water and complex marine environments. Given that research on the LWRs in the field of ocean engineering is not sufficiently abundant, the structural characteristics, hydrodynamic loads, global responses, fatigue damage assessment, and structural optimization progress of LWRs are systematically reviewed in this paper to provide references for researchers in related fields. Among these topics, the global response of LWRs is the main point of this review. This section details the theoretical analysis and numerical modeling methods employed in the study of LWRs’ global response, explores the research advancements in the vortex-induced vibration (VIV) related to LWRs, and discusses corresponding experimental studies. Finally, the installation, transfer, and repair processes of LWRs are investigated. Additionally, the importance of leveraging advanced technologies from other fields and combining them with current advanced algorithms is emphasized in efforts to assess fatigue damage and optimize the structures of LWRs, ultimately achieving complementary advantages.