  We explore the dynamics of stellar discs in the close vicinity of a
supermassive black hole (SMBH) by means of direct $N$-body simulations. We show
that an isolated nuclear stellar disc exhibits anisotropic mass segregation
meaning that massive stars settle to lower orbital inclinations and more
circular orbits than the light stars. However, in systems in which the stellar
disc is embedded in a much more massive isotropic stellar cluster, anisotropic
mass segregation tends to be suppressed. In both cases, an initially thin
stellar disc becomes thicker, especially in the inner parts due to the
fluctuating anisotropy in the spherical component. We find that vector resonant
relaxation is quenched in the disc by nodal precession, but it is still the
most efficient relaxation process around SMBHs of mass $10^6M_\odot$ and above.
Two body relaxation may dominate for less massive SMBHs found in dwarf
galaxies. Stellar discs embedded in massive isotropic stellar clusters
ultimately tend to become isotropic on the local two-body relaxation
time-scale. Our simulations show that the dynamics of young stars at the centre
of the Milky Way is mostly driven by vector resonant relaxation leading to an
anticorrelation between the scatter of orbital inclinations and distance from
the SMBH. If the $S$-stars formed in a disc less than 10 Myr ago, they may
coexist with a cusp of stellar mass black holes or an intermediate mass black
hole with mass up to $1000M_\odot$ to reproduce the observed scatter of angular
momenta.
