This volume is a basic introduction to certain aspects of elliptic functions and elliptic integrals that has evolved as part of the first semester of a course on theoretical and mathematical methods given to first- and second-year graduate students in physics and chemistry at the University of North Dakota. It introduces the subject as a moderate extension of ordinary trigonometry in which the reference circle is replaced by an ellipse. The complex analytic properties are introduced naturally so that a more complete study becomes possible. Elliptic Functions and Elliptic Integrals is the first volume to be published from the lecture series Lectures on Selected Topics in Mathematical Physics from William A Schwalm.