This book is about some of the ideas that are important for students of mathematics at the stage of transition from school to university. It contains 200 exercises on a wide range of interrelated topics in algebra, analysis and geometry, with explanations and informative comments. It has a comprehensive index, and can be used either as a reference book or as a framework for a systematic course of study. The book originated in sets of annotated exercises prepared for students and teachers working to an experimental syllabus of the Welsh Joint Education Committee. It will be useful to undergraduates and others who wish to develop their understanding of fundamental mathematics.