  For a Lie algebra $L$ and a subalgebra $M$ of $L$ we say that a subalgebra
$U$ of $L$ is a {\em supplement} to $M$ in $L$ if $L = M + U$. We investigate
those Lie algebras all of whose maximal subalgebras have abelian supplements,
those that have nilpotent supplements, those that have nil supplements, and
those that have supplements with the property that their derived algebra is
inside the maximal subalgebra being supplemented. For the algebras over an
algebraically closed field of characteristic zero in the last three of these
classes we find complete descriptions; for those in the first class partial
results are obtained.
