  Smooth Cauchy data for the Einstein-Lambda-vacuum field equations with
positive cosmological constant Lambda that are sufficiently close to de Sitter
data develop into a solution that admits a smooth conformal boundary Scri+ in
its future. The conformal Einstein equations determine a smooth conformal
extension across Scr+ that defines on `the other side' again a Lambda-vacuum
solution. In this article we discuss to what extent these properties generalize
to the future asymptotic behaviour of solutions to the Einstein-Lambda
equations with matter. We study FLRW solutions and the Einstein-Lambda
equations coupled to conformally covariant matter transport equations, to
conformally privileged matter equations, and to conformally non-covariant
matter equations. We present recent results on the
Einstein-Lambda-perfect-fluid equations with a non-linear asymptotic dust or
asymptotic radiation equation of state.
