  We discuss domain walls from spontaneous breaking of Abelian discrete
symmetries $Z_N$. A series of different domain wall structures are predicted,
depending on the symmetry and charge assignments of scalars leading to the
spontaneous symmetry breaking (SSB). A widely-existing type of domain walls are
those separating degenerate vacua which are adjacent in the field space. We
denote these walls as adjacency walls. In the case that $Z_N$ terms are small
compared with the $U(1)$ terms, the SSB of $U(1)$ generates strings first and
then adjacency walls bounded by strings are generated after the SSB of $Z_N$.
For symmetries larger than $Z_3$, non-adjacent vacua exist, we regard walls
separating them as non-adjacency walls. These walls are unstable if $U(1)$ is a
good approximation. If the discrete symmetry is broken via multiple steps, we
arrive at a complex structure that one kind of walls wrapped by another type.
On the other hand, if the symmetry is broken in different directions
independently, walls generated from the different breaking chains are blind to
each other.
