Sampling could be either done regularly (say along some discrete subgroup), or irregularly by random sampling. The sampling problem is now equivalent to a sampling problem on the unit circle. This is, in fact a compact manifold, which is also a Lie group. Since the theory of finite frames, often focuses its study on complex Hilbert spaces, this chapter provides focus to the action of the unitary group of matrices on C ⁿ . It discusses a number of ways in which, Lie theory gives frame theory a language, useful in organizing and categorizing “smoothly” all orthonormal bases, Riesz bases, n-Parseval frames and n-frames. The unitary group admits several pairs of matrices, each generating a free group. Since the harmonic analysis of such groups is generally quite poorly understood, it is best to replace any countable subgroup of the unitary group with its topological closure, which is of course a Lie group.