  NMR experiments on liquid $^3$He infused into uniaxially anisotropic silica
aerogels show the stabilisation of two equal-spin-pairing chiral phases on
cooling from the normal phase. The alignment of the chiral axis relative to the
anisotropy axis for these phases is predicted to depend upon temperature. A
chiral A-like phase is also stabilized when $^3$He is confined to a slab of
thickness $D\sim \xi$, the superfluid coherence length. For both types of
confinement, scattering of quasiparticles by the random potential - aerogel or
surface - is pair breaking and generates a sub-gap density of quasiparticle
states. The random field also conspires with the chiral order parameter to
generate skew scattering of quasiparticles in the plane normal to the chiral
axis. This scattering mechanism leads to anomalous thermal Hall transport for
nonequilibrium quasiparticles driven by a thermal gradient. We report
theoretical results for the anomalous thermal Hall conductivity for theoretical
models for chiral phases of $^3$He in both anisotropic aerogel and slabs. The
anomalous thermal Hall effect (ATHE) provides an important tool to identify
signatures of broken time-reversal and mirror symmetries and topology in chiral
superconductors/superfluids.
