  We compute the monodromy of the Hitchin fibration for the moduli space of
$L$-twisted $SL(n,\mathbb{C})$ and $GL(n,\mathbb{C})$-Higgs bundles for any
$n$, on a compact Riemann surface of genus $g>1$. We require the line bundle
$L$ to either be the canonical bundle or satisfy $deg(L) > 2g-2$. The monodromy
group is generated by Picard-Lefschetz transformations associated to vanishing
cycles of singular spectral curves. We construct such vanishing cycles
explicitly and use this to show that the $SL(n,\mathbb{C})$ monodromy group is
a {\em skew-symmetric vanishing lattice} in the sense of Janssen. Using the
classification of vanishing lattices over $\mathbb{Z}$, we completely determine
the structure of the monodromy groups of the $SL(n,\mathbb{C})$ and
$GL(n,\mathbb{C})$ Hitchin fibrations. As an application we determine the image
of the restriction map from the cohomology of the moduli space of Higgs bundles
to the cohomology of a non-singular fibre of the Hitchin fibration.
