  Insulating states can be topologically nontrivial, a well-established notion
that is exemplified by the quantum Hall effect and topological insulators. By
contrast, topological metals have not been experimentally evidenced until
recently. In systems with strong correlations, they have yet to be identified.
Heavy fermion semimetals are a prototype of strongly correlated systems and,
given their strong spin-orbit coupling, present a natural setting to make
progress. Here we advance a Weyl-Kondo semimetal phase in a periodic Anderson
model on a noncentrosymmetric lattice. The quasiparticles near the Weyl nodes
develop out of the Kondo effect, as do the surface states that feature Fermi
arcs. We determine the key signatures of this phase, which are realized in the
heavy fermion semimetal Ce$_3$Bi$_4$Pd$_3$. Our findings provide the
much-needed theoretical foundation for the experimental search of topological
metals with strong correlations, and open up a new avenue for systematic
studies of such quantum phases that naturally entangle multiple degrees of
freedom.
