We will present some results on the Laplacian flow of $G_2$-structures and its solitons in the homogeneous case, including the following:

Long time existence for any closed Laplacian flow solution in the context of solvable Lie groups with a codimension-one abelian normal subgroup.

Many examples of closed expanding Laplacian solitons which are not eigenfunctions.

First examples of closed Laplacian solitons which are shrinking, and in particular produce closed Laplacian flow solutions with a finite-time singularity.

Extremally Ricci pinched G2-structures (introduced by Bryant) which are steady Laplacian solitons.