  This paper introduces two new families of non-parametric tests of
goodness-of-fit on the compact classical groups. One of them is a family of
tests for the eigenvalue distribution induced by the uniform distribution,
which is consistent against all fixed alternatives. The other is a family of
tests for the uniform distribution on the entire group, which is again
consistent against all fixed alternatives. We find the asymptotic distribution
under the null and general alternatives. The tests are proved to be
asymptotically admissible. Local power is derived and the global properties of
the power function against local alternatives are explored.
  The new tests are validated on two random walks for which the mixing-time is
studied in the literature. The new tests, and several others, are applied to
the Markov chain sampler proposed by \cite{jones2011randomized}, providing
strong evidence supporting the claim that the sampler mixes quickly.
