  We study the existence problem for Novikov algebra structures on
finite-dimensional Lie algebras. We show that a Lie algebra admitting a Novikov
algebra is necessarily solvable. Conversely we present a $2$-step solvable Lie
algebra without any Novikov structure. We use extensions and classical
$r$-matrices to construct Novikov structures on certain classes of solvable Lie
algebras.
