We compare the maximal dimension of abelian subalgebras and the maximal dimension of abelian ideals for finite-dimensional Lie algebras. We show that these dimensions coincide for solvable Lie algebras over an algebraically closed field of characteristic zero. We compute this invariant for all complex nilpotent Lie algebras of dimension n less than 8. Furthermore we study the case where there exists an abelian subalgebra of codimension 2. Here we explicitly construct an abelian ideal of codimension 2 in case of nilpotent Lie algebras.