  We study almost inner derivations of Lie algebras, which were introduced by
Gordon and Wilson in their work on isospectral deformations of compact
solvmanifolds. We compute all almost inner derivations for low-dimensional Lie
algebras, and introduce the concept of fixed basis vectors for proving that all
almost inner derivations are inner for $2$-step nilpotent Lie algebras
determined by graphs, free $2$ and $3$-step nilpotent Lie algebras, free
metabelian nilpotent Lie algebras on two generators, almost abelian Lie
algebras and triangular Lie algebras. On the other hand we also exhibit
families of nilpotent Lie algebras having an arbitrary large space of non-inner
almost inner derivations.
