I first became enamoured of the Fels and Olver formulation of the moving frames theory when it helped me solve a problem I had been thinking about for several years. I set about reading their two 50-page papers, and made a 20-page handwritten glossary of definitions. I was lucky in that I was able to ask Peter Olver many questions and am eternally grateful for the answers.

I set about solving the problems that interested me, and realised there were so many of them that I could write a book. I also wanted to share my amazement at just how powerful the methods were, and at the essential simplicity of the central idea. What I have tried to achieve in this book is a discussion rich in examples, exercises and explanations that is largely accessible to a graduate student, although access to a professional mathematician will be required for some parts. I was extremely fortunate to have six students read through various drafts from the very beginning. The comments and hints they needed have been incorporated, and I have not hesitated to put in a discussion, example, exercise or hint that might be superfluous to a professional.

There is a fair amount of original material in this book. Even though some of the problems addressed here have been solved using moving frames already, I have re-proved some results to keep both solution methods and proofs within the domain of the mathematics developed here.