  We present a systematic study of how vortices in superfluid films interact
with the spatially varying Gaussian curvature of the underlying substrate. The
Gaussian curvature acts as a source for a geometric potential that attracts
(repels) vortices towards regions of negative (positive) Gaussian curvature
independently of the sign of their topological charge. Various experimental
tests involving rotating superfluid films and vortex pinning are first
discussed for films coating gently curved substrates that can be treated in
perturbation theory from flatness. An estimate of the experimental regimes of
interest is obtained by comparing the strength of the geometrical forces to the
vortex pinning induced by the varying thickness of the film which is in turn
caused by capillary effects and gravity. We then present a non-perturbative
technique based on conformal mappings that leads an exact solution for the
geometric potential as well as the geometric correction to the interaction
between vortices. The conformal mapping approach is illustrated by means of
explicit calculations of the geometric effects encountered in the study of some
strongly curved surfaces and by deriving universal bounds on their strength.
