Let $K$ be a number field and $E/K$ an elliptic curve without
complex multiplication. A well-known theorem of Serre asserts that the
Galois group of $K(E[\ell])/K$ is as all of ${\rm GL}_2(\Z/\ell)$ for any
sufficiently large prime $\ell$. If we replace $E/K$ by a polarized abelian
variety $A/K$ with trivial endomorphism ring, then Serre later showed
that the Galois group of $K(A[\ell])/K$ is also as large as possible, for
all sufficiently large $\ell$, provided $\dim(A)$ is 2,6 or odd. We will
show how to prove a similar result for `most' $A$ and without any
restriction on $\dim(A)$.