In this short review, we present the current status about the theoretical and experimental studies for some important semileptonic decays of mesons. We firstly gave a brief introduction for the experimental measurements for decays, the BaBar's and anomaly, the deviation for decay. We then made a careful discussion about the evaluations for the relevant form factors in the light-cone QCD sum rule, 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 mesons. We also made careful phenomenological analysis for these pQCD predictions and found, in general, the following points: (a) For all the considered semileptonic decays, the next-to-leading order pQCD predictions for their decay rates agree well with the data and those from other different theoretical methods; (b) For and , the pQCD predictions agree very well with the data, the BaBar's anomaly of are therefore explained successfully in the standard model by employing the pQCD approach; and (c) We defined several new ratios and , they may be more sensitive to the QCD dynamics which controls the transitions than the old ratios, we therefore strongly suggest LHCb and the forthcoming Super- experiments to measure these new ratios.