The importance of quality in software engineering is widely recognized today. As a result, agile methods have incorporated quality practices in the software development process to achieve quality software quickly and in harmony with rapidly changing customer requirements. Experts have published a substantial amount of information related to the diverse aspects of software quality and agile development. This work aims to perform a systematic literature review of the following characteristics: critical success factors, metrics to measure quality, quality attributes, agile practices, quality models, and agile principles considered by these models. 773 papers related to quality in the agile development process were identified, of which, for the preparation of this article, 71 papers were selected. This study catalogs 118 critical success factors, 137 agile practices, 102 metrics, and 70 quality attributes that affect the agile software development. In addition, 14 quality models, which do not cover all the aspects of quality related to agile development, were studied to prepare this paper. 28.17% of the studies analyzed consider the agile practices as they affect the quality of software, which is the most studied aspect.