It is quite easy to describe a Hadamard matrix. A (real) Hadamard matrix is an orthogonal matrix with entries which are all +-1. We will introduce some of the mathematical theory behind Hadamard matrices including the connection to group cohomology; however, only basic linear algebra and group theory will be necessary to follow the talk. Other topics covered will be frame theory, coding theory and cellular communications, as well as the application of Hadamard matrices to these areas.