  The $G$-strand equations for a map $\mathbb{R}\times \mathbb{R}$ into a Lie
group $G$ are associated to a $G$-invariant Lagrangian. The Lie group manifold
is also the configuration space for the Lagrangian. The $G$-strand itself is
the map $g(t,s): \mathbb{R}\times \mathbb{R}\to G$, where $t$ and $s$ are the
independent variables of the $G$-strand equations. The Euler-Poincar\'e
reduction of the variational principle leads to a formulation where the
dependent variables of the $G$-strand equations take values in the
corresponding Lie algebra $\mathfrak{g}$ and its co-algebra, $\mathfrak{g}^*$
with respect to the pairing provided by the variational derivatives of the
Lagrangian.
  We review examples of different $G$-strand constructions, including matrix
Lie groups and diffeomorphism group. In some cases the $G$-strand equations are
completely integrable 1+1 Hamiltonian systems that admit soliton solutions.
