  The intimate relations between Einstein's equation, conformal geometry,
geometric asymptotics, and the idea of an isolated system in general relativity
have been pointed out by Penrose many years ago. A detailed analysis of the
interplay of conformal geometry with Einstein's equation allowed us to deduce
from the conformal properties of the field equations a method to derive under
various assumptions definite statements about the feasibility of the idea of
geometric asymptotics.
  More recent investigations have demonstrated the possibility to analyse the
most delicate problem of the subject -- the behaviour of asymptotically flat
solutions to Einstein's equation in the region where ``null infinity meets
space-like infinity'' -- to an arbitrary precision. Moreover, we see now that
the, initially quite abstract, analysis yields methods for dealing with
practical issues. Numerical calculations of complete space-times in finite
grids without cut-offs become feasible now. Finally, already at this stage it
is seen that the completion of these investigations will lead to a
clarification and deeper understanding of the idea of an isolated system in
Einstein's theory of gravitation. In the following I wish to give a survey of
the circle of ideas outlined above, emphasizing the interdependence of the
structures and the naturalness of the concepts involved.
