In the past two decades there has been a considerable interest in
describing all possible gradings by abelian groups on simple Lie algebras. Over an
algebraically closed field of characteristic zero, the answer is nearly complete in
the finite-dimensional case: fine gradings have been classified up to equivalence
for all types, and arbitrary $G$-gradings, for a fixed group $G$, have been
classified up to isomorphism except for types $E_6$, $E_7$ and $E_8$. For a given
$G$-grading on a simple finite-dimensional Lie algebra $L$, we will discuss some
recent classification results for finite-dimensional graded $L$-modules.