The author proves a version of the Golod-Shafarevich theorem for a large class of Lie algebras, including all finite-dimensional and all soluble Lie algebras. Specifically, suppose that \(L\) is a finite-dimensional or soluble Lie algebra over a field \(k\) which has a presentation with \(n\) generators and \(r\) relations. Suppose also that \(d= \dim_k L/[ L,L ]\geq 2\). Then \[ r\geq n-d+ d^2/4. \] The analogous result is proven for \(L\) a finite-dimensional or soluble restricted Lie algebra over a field \(k\) of characteristic \(p\). Here \(d\) is the dimension of the largest abelian quotient of \(L\) with zero \(p\)-mapping, namely \(L/([ L,L ]+ L^p)\). In the restricted case there is a consequence for the structure of \(L\): there is a constant \(\kappa\) such that \[ \dim_k M/([ M,M ]+ M^p)\leq \kappa p^{(\dim_k L/M )/2} \] for each restricted subalgebra \(M\) of finite codimension in \(L\).