This book contains an informal but systematic account of the logical and algebraic foundations of mathematical analysis written at a fairly elementary level. The book is entirely self-contained but will be most useful to students who have already taken, or are in the process of taking, an introductory course in basic mathematical analysis. Such a course necessarily concentrates on the notion of convergence and the rudiments of the differential and integral calculus. Little time is therefore left for consideration of the foundations of the subject. But the foundational issues are too important to be neglected or to be left entirely in the hands of the algebraists (whose views on what is important do not always coincide with those of an analyst). In particular, a good grasp of the material treated in this book is essential as a basis for more advanced work in analysis. The fact remains, however, that a quart will not fit into a pint bottle and only so many topics can be covered in a given number of lectures. In my own lecture course I deal with this problem to some extent by encouraging students to read the more elementary material covered in this book for themselves, monitoring their progress through problem classes. This seems to work quite well and it is for this reason that substantial sections of the text have been written with a view to facilitating ‘self-study’, even though this leads to a certain amount of repetition and of discussion of topics which some readers will find very elementary. Readers are invited to skip rather briskly through these sections if at all possible.