  Thin $^4$He films adsorbed to weakly attractive substrates form nearly 2D
layers. We describe the vortices in 2D superfluid $^4$He like quasiparticles.
With the aid of a variational many-body calculation we estimate their inertial
mass and describe their interactions with the $^4$He particles and other
vortices. Third sound measurements revealed anomalous behavior below the
BKT-phase transition temperature. We ascribe this to the sound mode traveling
in the fluid of vortex-antivortex pairs. These pairs forms a crystal (or liquid
crystal) when the film thickness increases, the third sound mode splits into
two separate modes as seen in experiments. Our many-body calculation predicts
the critical density, at which the phase transition into the vortex-antivortex
state at zero temperature occurs. We also describe the phase diagram of thin
$^4$He films.
