The motivations for writing these notes arose while I was coteaching a seminar on Special Topics in Machine Perception with Kostas Daniilidis in the Spring of 2004. In the Spring of 2005, I gave a version of my course Advanced Geometric Methods in Computer Science (CIS610), with the main goal of discussing statistics on diffusion tensors and shape statistics in medical imaging. This is when I realized that it was necessary to cover some material on Riemannian geometry but I ran out of time after presenting Lie groups and never got around to doing it! Then, in the Fall of 2006 I went on a wonderful and very productive sabbatical year in Nicholas Ayache’s group (ACSEPIOS) at INRIA Sophia Antipolis whereI learned about the beautiful and exciting work of Vincent Arsigny, Olivier Clatz, Herve Delingette, Pierre Fillard, Gr ́goire Malandin, Xavier Pennec, Maxime Sermesant, and, of course, Nicholas Ayache, on statistics on manifolds and Lie groups applied to medical imaging. This inspired me to write chapters on differential geometry and, after a few additions made during Fall 2007 and Spring 2008, notably on left-invariant metrics on Lie groups, my little set of notes from 2004 had grown into the manuscript found here.