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In Memoriam Nikolai Uraltsev : Uraltsev's and other Sum Rules, Theory and Phenomenology of $D^{**}$'s

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 à 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.