  We study the second-order nonlinear optical response of a Weyl semimetal
(WSM), i.e. a three-dimensional metal with linear band touchings acting as
point-like sources of Berry curvature in momentum space, termed "Weyl-Berry
monopoles". We first show that the anomalous second-order photocurrent of WSMs
can be elegantly parametrized in terms of Weyl-Berry dipole and quadrupole
moments. We then calculate the corresponding charge and node conductivities of
WSMs with either broken time-reversal invariance or inversion symmetry. In
particular, we predict a universal dissipationless second-order anomalous node
conductivity for WSMs belonging to the TaAs family.
