There are many reasons for studying representation theory. In this lecture I will just touch on a few of them having to do with geometry and physics. We will start with the group SU(2) of 2×2 unitary matrices of determinant 1, and explain what its representations have to do with the spectrum of the Laplacian on the 2-sphere and 3-sphere, the spectrum of the hydrogen atom, and the spin of the electron. Then we will move on to applications of the representation theory of the "Heisenberg group" (the upper-triangular 3×3 real matrices with 1's on the diagonal).