  This paper studies twisted signature invariants and twisted linking forms,
with a view towards obstructions to knot concordance. Given a knot $K$ and a
representation $\rho$ of the knot group, we define a twisted signature function
$\sigma_{K,\rho} \colon S^1 \to \mathbb{Z}$. This invariant satisfies many of
the same algebraic properties as the classical Levine-Tristram signature
$\sigma_K$. When the representation is abelian, $\sigma_{K,\rho}$ recovers
$\sigma_K$, while for appropriate metabelian representations, $\sigma_{K,\rho}$
is closely related to the Casson-Gordon invariants. Additionally, we prove
satellite formulas for $\sigma_{K,\rho}$ and for twisted Blanchfield forms.
