Semileptonic heavy-to-heavy $B$ decays are very intriguing transitions because of the long-standing tension between the inclusive and the exclusive determinations of the Cabibbo-Kobayashi-Maskawa matrix element $\vert V_{cb} \vert$. Another discrepancy exists between the Standard Model (SM) expectations and the measurements of the ratios $R(D^{(*)})$, which are a test of Lepton Flavour Universality. In both cases, a central role is played by the hadronic Form Factors (FFs) describing these decays. My goal is to review the $b \to c$ quark transitions through the novel Dispersive Matrix approach, which is based on the non-perturbative determination of the dispersive bounds. I will describe how to obtain new values for $\vert V_{cb} \vert$ and $R(D^{(*)})$ from semileptonic $B \to D^{(*)}$ and $B_s \to D_s^{(*)}$ decays. Particular attention will be devoted to the semileptonic $B \to D^*$ channel, which is the main source of the aforementioned tensions. A critical comparison with the results from the Unitarity Triangle Analysis and the ones from HFLAV SM predictions will be then discussed. Finally, I will discuss in detail the state-of-the-art of the computations of the FFs on the lattice and their comparison with experimental measurements.
