This book is an alternative and highly engaging introduction to the highlights of a typical undergraduate mathematics course. Building on very simple principles, it develops these mathematical highlights, known to every well-rounded mathematician, in an intuitive and entertaining way. The aim of the book is to motivate and inspire the reader to discover and understand some of these truly amazing mathematical structures and ideas which are frequently not fully grasped, pass unnoticed or simply swamped in an undergraduate mathematics course. For the experienced mathematician the book offers refreshing, often enlightening, hindsight. For the novice it is an exciting intellectual journey.