  Discrete symmetries are widely imposed in particle theories. It is well-known
that the spontaneous breaking of discrete symmetries leads to domain walls.
Current studies of domain walls have focused on those from the spontaneous
breaking of a $Z_2$ symmetry. Larger discrete symmetries have multiple
degenerate vacua, leading to the domain walls in principle different from the
simplest $Z_2$ domain wall. We take domain walls from $Z_N$ symmetry breaking
as an illustrative study, and study in detail the $Z_3$ case, in which
semi-analytical results for the tension and thickness of domain walls are
derived. Explicit symmetry breaking terms lead to the dynamics of domain walls
collapsing more complicated than the $Z_2$ case. Gravitational wave signals
deviate from those from $Z_2$ domain walls.
