  We study balanced Hermitian structures on almost abelian Lie algebras, i.e.
on Lie algebras with a codimension-one abelian ideal. In particular, we
classify six-dimensional almost abelian Lie algebras which carry a balanced
structure. It has been conjectured by A. Fino and L. Vezzoni that a compact
complex manifold admitting both a balanced metric and a SKT metric necessarily
has a K\"ahler metric: we prove this conjecture for compact almost abelian
solvmanifolds with left-invariant complex structures. Moreover, we investigate
the behaviour of the flow of balanced metrics introduced by L. Bedulli and L.
Vezzoni and of the anomaly flow by D. H. Phong, S. Picard and X. Zhang on
almost abelian Lie groups. In particular, we show that the anomaly flow
preserves the balanced condition and that locally conformally K\"ahler metrics
are fixed points.
