  We calculate the shear viscosity $\eta = \eta_{e\mu}+\eta_{n}$ in a neutron
star core composed of nucleons, electrons and muons ($\eta_{e\mu}$ being the
electron-muon viscosity, mediated by collisions of electrons and muons with
charged particles, and $\eta_{n}$ the neutron viscosity, mediated by
neutron-neutron and neutron-proton collisions). Deriving $\eta_{e\mu}$, we take
into account the Landau damping in collisions of electrons and muons with
charged particles via the exchange of transverse plasmons. It lowers
$\eta_{e\mu}$ and leads to the non-standard temperature behavior
$\eta_{e\mu}\propto T^{-5/3}$. The viscosity $\eta_{n}$ is calculated taking
into account that in-medium effects modify nucleon effective masses in dense
matter. Both viscosities, $\eta_{e\mu}$ and $\eta_{n}$, can be important, and
both are calculated including the effects of proton superfluidity. They are
presented in the form valid for any equation of state of nucleon dense matter.
We analyze the density and temperature dependence of $\eta$ for different
equations of state in neutron star cores, and compare $\eta$ with the bulk
viscosity in the core and with the shear viscosity in the crust.
