  We develop a general framework for the evaluation of $d$-dimensional cut
Feynman integrals based on the Baikov-Lee representation of purely-virtual
Feynman integrals. We implement the generalized Cutkosky cutting rule using
Cauchy's residue theorem and identify a set of constraints which determine the
integration domain. The method applies equally well to Feynman integrals with a
unitarity cut in a single kinematic channel and to maximally-cut Feynman
integrals. Our cut Baikov-Lee representation reproduces the expected relation
between cuts and discontinuities in a given kinematic channel and furthermore
makes the dependence on the kinematic variables manifest from the beginning. By
combining the Baikov-Lee representation of maximally-cut Feynman integrals and
the properties of periods of algebraic curves, we are able to obtain complete
solution sets for the homogeneous differential equations satisfied by Feynman
integrals which go beyond multiple polylogarithms. We apply our formalism to
the direct evaluation of a number of interesting cut Feynman integrals.
