Lately, a new generation of experts, some of whom are very powerful researchers, has started disseminating exposition via interactive blogs. Since they receive immediate and continuous international feedback, the interests and needs of their audience inevitably inform the communication. Tao, who---even among Fields medalists---has exceptional intellectual range, writes among the most famous of these blogs, and the present volume contains a selection of redacted content from it. Whereas many textbooks aim to teach lower-division mathematics majors to prove trivial theorems, this book explores very generally how research mathematicians prove nontrivial theorems. But unifying themes aside, Tao has chosen very well as case studies sundry topics from mutually distant branches of mathematics. Those sufficiently elementary demand a healthy stretch, and none will seem overly familiar, even to most experts. The class exhortation 'read the masters' takes on new urgency in an era when the masters return favor. Highly recommended ... [for] all academic and professional library collections.