  In this paper, we use `generalized Seifert surfaces' to extend the
Levine-Tristram signature to colored links in S^3. This yields an integral
valued function on the m-dimensional torus, where m is the number of colors of
the link. The case m=1 corresponds to the Levine-Tristram signature. We show
that many remarkable properties of the latter invariant extend to this
m-variable generalization: it vanishes for achiral colored links, it is
`piecewise continuous', and the places of the jumps are determined by the
Alexander invariants of the colored link. Using a 4-dimensional interpretation
and the Atiyah-Singer G-signature theorem, we also prove that this signature is
invariant by colored concordance, and that it provides a lower bound for the
`slice genus' of the colored link.
