  Three-dimensional topological semimetals exhibit linear energy band crossing
points that act as monopoles of Berry curvature. Here, an alternative class of
semimetals is introduced, featuring linear $N$-fold crossing points each of
which acts as a source of a \emph{Berry dipole}. We construct continuum and
lattice models for such \emph{massless multifold Hopf semimetals (MMHSs)} with
$N=3,4,5$ bands and study nontrivial effects of a Berry dipole crossing: (i) A
Landau level spectrum that is strongly tunable by the orientation of the
magnetic field relative to the dipole axis. (ii) An anomalous Hall conductivity
that is an odd function of the Fermi level. (iii) Weak-field dissipative
magnetoconductivities that resemble the chiral anomaly, chiral magnetic and
magnetochiral effects familiar from a pair of coupled Weyl nodes, but that are
even functions of the Fermi level. By gapping out MMHSs, multiband Hopf
insulators with Hopf numbers as high as $\mathcal{N}_\text{Hopf}=10$ are
obtained, providing a fertile playground to explore delicate topology.
