Screen content, which is often computer-generated, has many characteristics distinctly different from conventional camera-captured natural scene content. Such characteristic differences impose major challenges to the corresponding content quality assessment, which plays a critical role to ensure and improve the final user-perceived quality of experience (QoE) in various screen content communication and networking systems. Quality assessment of such screen content has attracted much attention recently, primarily because the screen content grows explosively due to the prevalence of cloud and remote computing applications in recent years, and due to the fact that conventional quality assessment methods can not handle such content effectively. As the most technology-oriented part of QoE modeling, image/video content/media quality assessment has drawn wide attention from researchers, and a large amount of work has been carried out to tackle the problem of screen content quality assessment. This article is intended to provide a systematic and timely review on this emerging research field, including (1) background of natural scene vs. screen content quality assessment; (2) characteristics of natural scene vs. screen content; (3) overview of screen content quality assessment methodologies and measures; (4) relevant benchmarks and comprehensive evaluation of the state-of-the-art; (5) discussions on generalizations from screen content quality assessment to QoE assessment, and other techniques beyond QoE assessment; and (6) unresolved challenges and promising future research directions. Throughout this article, we focus on the differences and similarities between screen content and conventional natural scene content. We expect that this review article shall provide readers with an overview of the background, history, recent progress, and future of the emerging screen content quality assessment research.