  Domain wall solitons are basic constructs realizing phase transitions in
various field-theoretical models and are solutions to some nonlinear ordinary
differential equations descending from the corresponding full sets of governing
equations in higher dimensions. In this paper, we present a series of domain
wall solitons arising in several classical gauge field theory models. In the
context of the Abelian gauge field theory, we unveil the surprising result that
the solutions may explicitly be constructed, which enriches our knowledge on
integrability of the planar Liouville type equations in their one-dimensional
limits. In the context of the non-Abelian gauge field theory, we obtain some
existence theorems for domain wall solutions arising in the electroweak type
theories by developing some methods of calculus of variations formulated as
direct and constrained minimization problems over a weighted Sobolev space.
