This work provides a first taste of the theory of Lie groups accessible to advanced mathematics undergraduates and beginning graduate students, providing an appetiser for a more substantial further course. Although the formal prerequisites are kept as low level as possible, the subject matter is sophisticated and contains many of the key themes of the fully developed theory. We concentrate on matrix groups, i.e., closed subgroups of real and complex general linear groups. One of the results proved is that every matrix group is in fact a Lie group, the proof following that in the expository paper of Howe. Indeed, the latter, together with the book of Curtis, influenced our choice of goals for the present book and the course which it evolved from. As pointed out by Howe, Lie theoretic ideas lie at the heart of much of standard undergraduate linear algebra, and exposure to them can inform or motivate the study of the latter; we frequently describe such topics in enough detail to provide the necessary background for the benefit of readers unfamiliar with them.