  Discrete symmetries play an important role in several extensions of the
Standard Model (SM) of particle physics. For instance, in order to avoid flavor
changing neutral currents, a discrete $Z_2$ symmetry is imposed on the
Two-Higgs-Doublet Model (2HDM). This can lead to the formation of domain walls
(DW) as the $Z_2$ symmetry gets spontaneously broken during electroweak
symmetry breaking in the early universe and domain walls form between regions
whose vacua are related by the discrete symmetry. Due to this simultaneous
spontaneous breaking of both the discrete symmetry and the electroweak
symmetry, the vacuum manifold consists of two disconnected 3-spheres. Such a
non-trivial disconnected vacuum manifold leads to several choices for the vacua
at two adjacent regions, in contrast to models where only the discrete symmetry
gets spontaneously broken and the vacuum manifold consists of several
disconnected points. Due to this, we end up with several classes of DW
solutions having different properties localized inside the wall, such as charge
and/or CP violating vacua. We discuss the properties of these different classes
of DW solutions as well as the interaction of SM fermions with such topological
defects leading to different exotic phenomena such as, for example, the top
quark being transmitted or reflected off the wall as a bottom quark.
