  The multiferroic ferrimagnet Cu$_2$OSeO$_3$ with a chiral crystal structure
attracted a lot of recent attention due to the emergence of magnetic skyrmion
order in this material. Here, the topological properties of its magnon
excitations are systematically investigated by linear spin-wave theory and
inelastic neutron scattering. When considering Heisenberg exchange interactions
only, two degenerate Weyl magnon nodes with topological charges $\pm$2 are
observed at high-symmetry points. Each Weyl point splits into two as the
symmetry of the system is further reduced by including into consideration the
nearest-neighbor Dzyaloshinsky-Moriya interaction, crucial for obtaining an
accurate fit to the experimental spin-wave spectrum. The predicted topological
properties are verified by surface state and Chern number analysis.
Additionally, we predict that a measurable thermal Hall conductivity can be
associated with the emergence of the Weyl points, the position of which can be
tuned by changing the crystal symmetry of the material.
