We will survey different methods of proving functional inequalities for hypoelliptic diffusions and the corresponding heat kernels. Namely, some of these methods rely on the generalized curvature-dimension inequalities (due to Baudoin-Garofalo), and some use structure theory and a Fourier transform on Lie groups. For nilpotent groups, the latter can be interpreted as an eigenfunction expansion of the hypoelliptic heat kernel. This is based on joint work with S. Albeverio, M. Asaad, F. Baudoin, B. Driver, T. Melcher.