Context: With the rising complexity and scale of software systems, there is an ever-increasing demand for sophisticated and cost-effective software testing. To meet such a demand, there is a need for a highly-skilled software testing work-force (test engineers) in the industry. To address that need, many university educators worldwide have included software-testing education in their software engineering (SE) or computer science (CS) programs. Objective: Our objective in this paper is to summarize the body of experience and knowledge in the area of software-testing education to benefit the readers (both educators and researchers) in designing and delivering software testing courses in university settings, and to also conduct further education research in this area. Method: To address the above need, we conducted a systematic literature mapping (SLM) to synthesize what the community of educators have published on this topic. After compiling a candidate pool of 307 papers, and applying a set of inclusion/exclusion criteria, our final pool included 204 papers published between 1992 and 2019. Results: The topic of software-testing education is becoming more active, as we can see by the increasing number of papers. Many pedagogical approaches (how to best teach testing), course-ware, and specific tools for testing education have been proposed. Many challenges in testing education and insights on how to overcome those challenges have been proposed. Conclusion: This paper provides educators and researchers with a classification of existing studies within software-testing education. We further synthesize challenges and insights reported when teaching software testing. The paper also provides a reference ("index") to the vast body of knowledge and experience on teaching software testing.