Random matrices are cool! In this book we give an idea why random matrices are so fascinating and have become a centrepiece of modern mathematics. This an introduction to random matrix theory, giving an impression of some of the most important aspects of this modern subject. In particular, it covers the basic combinatorial and analytic theory around Wigner’s semicircle law, featuring also concentration phenomena, and the Tracy–Widom distribution of the largest eigenvalue. The circular law and a discussion of Voiculescu’s multivariate extension of the semicircle law, as an appetiser for free probability theory, also make an appearance. This book is based on a lecture series for graduate and advanced undergraduate students at Saarland University, Germany.