  The same physical reality in Two Higgs doublet model (2HDM) can be described
by different Lagrangians. We call this property the reparametrization
invariance (in space of Lagrangians) and study corresponding symmetry group and
its subgroup describing rephasing invariance. Next we consider the
$Z_2$-symmetry of the Lagrangian, which prevents a
$\phi_1\leftrightarrow\phi_2$ transitions, and the different levels of its
violation, soft and hard. We argue that the 2HDM with a soft breaking of
$Z_2$-symmetry is a natural model in the description of EWSB. We also consider
vacuum structure of the 2HDM. We find very simple condition for a CP violation
in the Higgs sector. In the Model II for Yukawa interactions we obtain the set
of relations among the couplings to gauge bosons and to fermions which allows
one to analyze different physical situations (including CP violation) in terms
of these very couplings, instead of the parameters of Lagrangian. We discuss
possible interaction of Higgs fields of SM or 2HDM with inflatory Higgs field
describing exponential expansion of Universe after Big Bang and possible
variation in the scenario of beginning of Time.
