  We discuss the Type-X (lepton-specific) two Higgs doublet model as a solution
of the anomaly of the muon $g-2$. We consider various experimental constraints
on the parameter space such as direct searches for extra Higgs bosons at the
LEP II and the LHC Run-I, electroweak precision observables, the decay of $B_s
\to \mu^+\mu^-$, and the leptonic decay of the tau lepton. We find that the
measurement of the tau decay provides the most important constraint, which
excludes the parameter region that can explain the muon $g-2$ anomaly at the
1$\sigma$ level. We then discuss the phenomenology of extra Higgs bosons and
the standard model-like Higgs boson ($h$) to probe the scenario favored by the
$g-2$ data at the collider experiments. We find that the $4\tau$, $3\tau$ and
$4\tau + W/Z$ signatures are expected as the main signal of the extra Higgs
bosons at the LHC. In addition, we clarify that the value of the $h\tau\tau$
coupling is predicted to be the standard model value times about $-1.6$ to
$-1.0$, and the branching fraction of the $h\to \gamma\gamma$ mode deviates
from the standard model prediction by $-30\%$ to $-15\%$. Furthermore, we find
that the exotic decay mode, $h$ decaying into the $Z$ boson and a light CP-odd
scalar boson, is allowed, and its branching fraction can be a few percent.
These deviations in the property of $h$ will be tested by the precision
measurements at future collider experiments.
