DevOps is an emerging paradigm that reduces the barriers between developers and operations teams to offer continuous fast delivery and enable quick responses to changing requirements within the software life cycle. A significant volume of activity has been carried out in recent years with the aim of coupling DevOps stages with tools and methods to improve the quality of the produced software and the underpinning delivery methodology. While the research community has produced a sustained effort by conducting numerous studies and innovative development tools to support quality analyses within DevOps, there is still a limited cohesion between the research themes in this domain and a shortage of surveys that holistically examine quality engineering work within DevOps. In this paper, we address the gap by comprehensively surveying existing efforts in this area, categorizing them according to the stage of the DevOps lifecycle to which they primarily contribute. The survey holistically spans across all the DevOps stages, identify research efforts to improve architectural design, modeling and infrastructure-as-code, continuous-integration/continuous-delivery (CI/CD), testing and verification, and runtime management. Our analysis also outlines possible directions for future work in quality-aware DevOps, looking in particular at AI for DevOps and DevOps for AI software .