  In several models of beyond Standard Model physics (BSM) discrete symmetries
play an important role. For instance, in order to avoid flavor changing neutral
currents (FCNC), a discrete $Z_2$ symmetry is imposed on
Two-Higgs-Doublet-Models (2HDM). This can lead to the formation of domain walls
(DW) as the $Z_2$ symmetry gets spontaneously broken during electroweak
symmetry breaking (EWSB) in the early universe. Due to this simultaneous
spontaneous breaking of both the discrete symmetry and the electroweak
symmetry, the vacuum manifold has the structure of two disconnected 3-spheres
and the formed domain walls can exhibit several special properties in contrast
to standard domain walls. We focus on some of these properties such as CP and
electric charge violating vacua localized inside the domain walls. The breaking
of $U(1)_{em}$ inside the wall leads to the known phenomenon of
"clash-of-symmetries" mechanism, meaning that the symmetry group inside the
wall is smaller than the symmetry group far from the wall. We also discuss the
scattering of top quarks off such types of domain walls and show, for example,
that they can be reflected or transmitted off the wall as a bottom quark.
