A $G$-strand is a map $\mathbb{R}\times\mathbb{R}\to G$ into a Lie group $G$ that follows from Hamilton's principle for a certain class of $G$-invariant Lagrangians. $G$-strands on finite-dimensional groups satisfy $1+1$ space-time evolutionary equations. A large class of these equations have Lax-pair representations that show they admit soliton solutions. For example, the $SO(3)$-strand equations may be regarded physically as integrable dynamics for solitons on a continuous spin chain. Various other examples will be discussed, including collisions of solutions with singular support (e.g., peakons) on ${\rm Diff}(\mathbb{R})$-strands, in which ${\rm Diff}(\mathbb{R})$ is the group of diffeomorphisms of the real line $\mathbb{R}$, for which the group product is composition of smooth invertible functions.