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 6-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 Kahler 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 of 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 Kahler metrics are fixed points.
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