  In the alignment limit of a multi-doublet Higgs sector, one of the Higgs mass
eigenstates aligns with the direction of the scalar field vacuum expectation
values, and its couplings approach those of the Standard Model (SM) Higgs
boson. We consider CP-conserving Two-Higgs-Doublet Models (2HDMs) of Type I and
Type II near the alignment limit in which the lighter of the two CP-even Higgs
bosons, $h$, is the SM-like state observed at 125 GeV. In particular, we focus
on the 2HDM parameter regime where the coupling of $h$ to gauge bosons
approaches that of the SM. We review the theoretical structure and analyze the
phenomenological implications of the regime of alignment limit without
decoupling, in which the other Higgs scalar masses are not significantly larger
than $m_h$ and thus do not decouple from the effective theory at the
electroweak scale. For the numerical analysis, we perform scans of the 2HDM
parameter space employing the software packages 2HDMC and Lilith, taking into
account all relevant pre-LHC constraints, the latest constraints from the
measurements of the 125 GeV Higgs signal at the LHC, as well as the most recent
limits coming from searches for heavy Higgs-like states. We contrast these
results with the alignment limit achieved via the decoupling of heavier scalar
states, where $h$ is the only light Higgs scalar. Implications for Run 2 at the
LHC, including expectations for observing the other scalar states, are also
discussed.
