  We review recent results on the Bethe Ansatz solutions for the eigenvalues of
the transfer matrix of an integrable open XXZ quantum spin chain using
functional relations which the transfer matrix obeys at roots of unity. First,
we consider a case where at most two of the boundary parameters
{{$\alpha_-$,$\alpha_+$,$\beta_-$,$\beta_+$}} are nonzero. A generalization of
the Baxter $T-Q$ equation that involves more than one independent $Q$ is
described. We use this solution to compute the boundary energy of the chain in
the thermodynamic limit. We conclude the paper with a review of some results
for the general integrable boundary terms, where all six boundary parameters
are arbitrary.
