ABSTRACT:
The simplest model for Riemannian geometry is Euclidean space
R^n.  Analogously, the simplest model for a
Carnot--Caratheodory geometry is a nilpotent Lie group $N$ say,
with a few additional properties.  The simplest model for a
parabolic geometry is a nilpotent Lie group with somewhat
different properties.  This talk is a survey of various results
on the differential geometry of nilpotent Lie groups.  Several
notions of conformality and quasiconformality are described,
together with related regularity and rigidity theorems.