Abstract: Given a commutative ring R, we can define the unitary Cayley graph of R as the graph with vertices labeled by the elements of R, with x adjacent to y if and only if x-y is a unit in R. These graphs are full of symmetry and structure--which often makes computing graph parameters for them quite nice. We will talk about why these graphs are important, investigate some connections to number theory, and then play around with various graph parameters. This talk should require no background to understand and will end with some fun problems to try.