This handbook documents the QUASAR (QUality Assessment of System ARchitectures) method for assessing the quality of the architecture of a software-intensive system. It begins by discussing the challenges that are faced when assessing a system's architecture and outlines the development history of the method. The next section of the handbook documents the concept of quality cases and the claims, arguments, and evidence that compose them. This is followed by a description of the teams that collaborate to perform QUASAR tasks. Next, individual tasks and associated steps performed as part of the QUASAR method are documented. Next, the work products produced by these teams when performing these tasks are described. Finally, lessons learned during the development and use of the method when assessing the quality of major subsystems during the development of a very large, software-intensive system of systems are presented. Also provided are appendices that define common quality factors and subfactors, offer reusable checklists, and give examples of quality cases. The example quality cases illustrate valid quality goals and requirements that compose claims, example architecture decisions and associated rationales that compose arguments, and the types of evidence that architects might provide.