  The classical Waring problem deals with expressing every natural number as a
sum of g(k) kth powers. Similar problems for finite simple groups have been
studied recently, and in this paper we study them for finite quasisimple groups
G.
  We show that for a fixed non-trivial group word w and large enough G we have
w(G)^3=G, namely every element of G is a product of 3 values of w. For various
families of finite quasisimple groups, including covers of alternating groups,
we obtain a stronger result, namely w(G)^2=G. However, in contrast with the
case of simple groups, we show that w(G)^2 need not equal G for all large G. If
k>2 then x^k y^k fails to be surjective for infinitely many finite quasisimple
groups.
  The case k=2 turns out to be exceptional. Indeed, our last result shows that
every element of a finite quasisimple group is a product of two squares. This
can be regarded as a non-commutative analogue of Lagrange's four squares
theorem.
