These notes are intended for a general mathematical audience. In particular, we have in mind that they could be used as a course for undergraduates. They contain an explicit construction of highly connected but sparse graphs known as expander graphs. Besides their interest in combinatorics and graph theory, these graphs have applications to computer science and engineering. Our aim has been to give a self-contained treatment. Thus, the relevant background material in graph theory, number theory, group theory, and representation theory is presented. The text can be used as a brief introduction to these modern subjects as well as an example of how such topics are synthesized in modern mathematics. Prerequisites include linear algebra together with elementary algebra, analysis, and combinatorics.