  We first discuss Uraltsev's and other sum rules constraining the $B \to
D^{**}(L=1)$ weak transitions in the infinite mass limit, and compare them with
dynamical approaches in the same limit. After recalling these well established
facts, we discuss how to apply infinite mass limit to the physical situation.
We provide predictions concerning semi-leptonic decays and non-leptonic ones,
based on quark models. We then present in more detail the dynamical approaches:
the relativistic quark model \`a la Bakamjian-Thomas and lattice QCD. We
summarise lattice QCD results in the infinite mass limit and compare them to
the quark model predictions. We then present preliminary lattice QCD results
with finite $b$ and $c$ quark masses. A systematic comparison between theory
and experiment is performed. We show that some large discrepancies exist
between different experiments. Altogether the predictions at infinite mass are
in fair agreement with experiment for non-leptonic decays contrary to what
happens for semileptonic decays. We conclude by considering the prospects to
clarify both the experimental situation, the theoretical one and the comparison
between both.
