Updating various theoretical and experimental constraints on the four different types of two-Higgsdoublet models (2HDMs), we find that only the ‘lepton-specific’ (or ‘type-X’) 2HDM can explain the present muon $g−2$ anomaly in the parameter region of large tan $\beta$, a light CP-odd boson, and heavier CP-even and charged bosons which are almost degenerate. Severe constraints on the models come mainly from the consideration of vacuum stability and perturbativity, the electroweak precision data, the b-quark observables like $B_{S}\rightarrow \mu\mu$, the precision measurements of the lepton universality as well as the 125 GeV boson property observed at the LHC.