  An intrinsic definition in terms of conformal capacity is proposed for the
conformal type of a Carnot--Carath\'eodory space (parabolic or hyperbolic).
Geometric criteria of conformal type are presented. They are closely related to
the asymptotic geometry of the space at infinity and expressed in terms of the
isoperimetric function and the growth of the area of geodesic spheres. In
particular, it is proved that a sub-Riemannian manifold admits a conformal
change of metric that makes it into a complete manifold of finite volume if and
only if the manifold is of conformally parabolic type. Further applications are
discussed, such as the relation between local and global invertibility
properties of quasiconformal immersions (the global homeomorphism theorem).
