Abstract: We are going to discuss some recent results pertaining to the existence and non-existence of exponential Gabor bases and frames. In particular, we are going to prove that there is no orthogonal Gabor basis with a window function given by the characteristic function of the ball. We will also prove that if $sigma$ is the surface measure on the sphere $S^{d-1}$, then $L^2(sigma)$ does not possess a frame of exponentials. The interaction of analytic and combinatorial ideas will be emphasized throughout.