The first three editions of this popular textbook attracted a loyal readership and widespread use. Students find the book to be concise, accessible, and complete. Instructors find the book to be clear, authoritative, and dependable. The goal of this new edition is to make real analysis relevant and accessible to a broad audience of students with diverse backgrounds. Real analysis is a basic tool for all mathematical scientists, ranging from mathematicians to physicists to engineers to researchers in the medical profession. This text aims to be the generational touchstone for the subject and the go-to text for developing young scientists. In this new edition we endeavor to make the book accessible to a broader audience. This edition includes more explanation, more elementary examples, and the author stepladders the exercises. Figures are updated and clarified. We make the sections more concise, and omit overly technical details. We have updated and augmented the multivariable material in order to bring out the geometric nature of the topic. The figures are thus enhanced and fleshed out. Features A renewed enthusiasm for the topic comes through in a revised presentation A new organization removes some advanced topics and retains related ones Exercises are more tiered, offering a more accessible course Key sections are revised for more brevity