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After teaching mathematical statistics for several years using chalk on a blackboard (and, later, smelly “dry erase markers” on a whiteboard) mostly doing proofs of theorems, I decided to lecture from computer slides that provide an outline of the “big picture”. Rather than spend class time “doing” proofs that are given in standard texts, I decided that time would be better spent discussing the material from a different, higher-level perspective. While lecturing from canned slides, I cannot, however, ignore certain details of proofs and minutiae of examples. But what details and which minutiae? To be effective, my approach depends on an understanding between students and the instructor, an understanding that is possibly implicit. I lecture; but I ask “what is … ?” and “why is … ?”; and I encourage students to ask “what is … ?” and “why is … ?”. I adopt the attitude that there are many things that I don’t know, but if there’s something that I wonder about, I’ll admit ignorance and pursue the problem until I’ve attained some resolution. I encourage my students to adopt a similar attitude. I am completely dissatisfied with a class that sits like stumps on a log when I ask “what is … ?” or “why is … ?” during a lecture. What can I say? After writing class slides (in LATEX2″, of course), mostly in bullet form, I began writing text around the bullets, and I put the notes on the class website. Later I decided that a single document would be useful to serve as a companion for the study of mathematical statistics, especially if it had a comprehensive subject index, and extensive cross refererences within the document. In the electronic form, both the subject index and the internal cross references have active links to the appropriate pages in the book. This is not something that can be done with a hardcopy book. (One thing I must warn you about is that there is a (known) bug in the LATEX package hyperref; if the referenced point happens to occur at the top of a page, the link takes you to the previous page – so if you don’t see what you expect, try going to the next page.) Much of the present document reflects its origin as classroom notes; it contains incomplete sentences or sentence fragments, and it lacks connective material in some places. (The connective material was (probably!) supplied orally during the lectures.) Another characteristic of this document that results from the nature of its origin, as well as from the medium itself (electronic) is its length. A long book doesn’t use any more paper or “kill any more trees” than a short book. Usually, however, we can expect the density of “importance” in a short document to be greater than that in a long document. If I had more time I could make this book shorter by prioritizing its content, and I may do that someday. Several sections are incomplete and several proofs are omitted. Also, I plan to add more examples. I just have not had time to type up the material. I do not recommend that you print these notes. First of all, they are evolving, so a printed version is just a snapshot. Secondly, the PDF file contains active internal links, so navigation is easy. (For versions without active links, I try to be friendly to the reader by providing page numbers with most internal references.) This document is directed toward students for whom the theory of statistics is or will become an important part of their lives. Obviously, such students should be able to work through the details of “hard” proofs and derivations; that is, students should master the fundamentals of mathematical statistics. In addition, students at this level should acquire, or begin acquiring, a deep appreciation for the field, including its historical development and its relation to other areas of mathematics and science generally; that is, students should master the fundamentals of the broader theory of statistics. Some of the chapter endnotes are intended to help students gain such an appreciation by leading them to background sources and also by making more subjective statements than might be made in the main body.