  An almost abelian Lie group is a solvable Lie group with a codimension-one
normal abelian subgroup. We characterize almost Hermitian structures on almost
abelian Lie groups where the almost complex structure is harmonic with respect
to the Hermitian metric. Also, we adapt the Gray-Hervella classification of
almost Hermitian structures to the family of almost abelian Lie groups. We
provide several examples of harmonic almost complex structures in different
Gray-Hervella classes on some associated compact almost abelian solvmanifolds.
