  In this short review, we present the current status about the theoretical and
experimental studies for some important semileptonic decays of $B/B_s$ mesons.
We firstly gave a brief introduction for the experimental measurements for
$B/B_s \to P (l^+l^-, l^-\bar{\nu}_l, \nu \bar{\nu})$ decays, the BaBar's
$R(D)$ and $R(D^*)$ anomaly, the $P_5^\prime$ deviation for $B^0 \to K^{*0}
\mu^+ \mu^-$ decay. We then made a careful discussion about the evaluations for
the relevant form factors in the light-cone QCD sum rule (LCSRs), the heavy
quark effective theory, and the perturbative QCD factorization approach. By
using the form factors calculated in the perturbative (pQCD) approach, we then
calculate and show the pQCD predictions for the decay rates of many
semileptonic decays of $B/B_s$ mesons. We also made careful phenomenological
analysis for these pQCD predictions and found, in general, the following
points: (a) For all the considered $B/B_s$ semileptonic decays, the
next-to-leading order (NLO) pQCD predictions for their decay rates agree well
with the data and those from other different theoretical methods; (b) For
$R(D)$ and $R(D^*)$, the pQCD predictions agree very well with the data, the
BaBar's anomaly of $R(D^{(*)})$ are therefore explained successfully in the
standard model by employing the pQCD approach; and (c) We defined several new
ratios $R_D^{l,\tau}$ and $R_{D_s}^{l,\tau}$, they may be more sensitive to the
QCD dynamics which controls the $B/B_s \to (D^{(*)},D_s^{(*)})$ transitions
than the old ratios, we therefore strongly suggest LHCb and the forthcoming
Super-B experiments to measure these new ratios.
