The aim of this course is to introduce a variety of areas of mathematics and their applications in Physical Chemistry. The course will be at a level appropriate to a beginning graduate student in chemistry who has taken two years of undergraduate mathematics courses. The course will develop fundamental concepts taken from complex variables, methods for the solution of ordinary differential equations, and partial differential equations. Throughout the course, we illustrate utility of the concepts to various spectroscopies, including electronic and optical. An introduction to the required physical concepts in electronics and optics will be developed as needed.