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.