Context: Quality of Service (QoS) is a major issue in various web service related activities. Quality models have been proposed as the engineering artefact to provide a common framework of understanding for QoS, by defining the quality factors that apply to web service usage. Objective: The goal of this study is to evaluate the current state of the art of the proposed quality models for web services, specifically: (1) which are these proposals and how are they related; (2) what are their structural characteristics; (3) what quality factors are the most and least addressed; and (4) what are their most consolidated definitions. Method: We have conducted a systematic mapping by defining a robust protocol that combines automatic and manual searches from different sources. We used a rigorous method to elicitate the keywords from the research questions and a selection criteria to retrieve the final papers to evaluate. We have adopted the ISO/IEC 25010 standard to articulate our analysis. Results: We have evaluated 47 different quality models from 65 papers that fulfilled the selection criteria. By analyzing in depth these quality models, we have: 1) distributed the proposals along the time dimension and identified their relationships; 2) analysed their size (visualizing the number of nodes and levels) and definition coverage (as indicator of quality of the proposals); 3) quantified the coverage of the different ISO/IEC 25010 quality factors by the proposals; 4) identified the quality factors that appeared in at least 30% of the surveyed proposals and provided the most consolidated definitions for them. Conclusions: We believe that this panoramic view on the anatomy of the quality models for web services may be a good reference for prospective researchers and practitioners in the field and especially may help avoiding the definition of new proposals that do not align with current research.