  We study ideals of Novikov algebras and Novikov structures on
finite-dimensional Lie algebras. We present the first example of a three-step
nilpotent Lie algebra which does not admit a Novikov structure. On the other
hand we show that any free three-step nilpotent Lie algebra admits a Novikov
structure. We study the existence question also for Lie algebras of triangular
matrices. Finally we show that there are families of Lie algebras of arbitrary
high solvability class which admit Novikov structures.
