When he first introduced the notion of a conformal boundary into the study of asymptotically empty space–times, Penrose noted that the boundary would be null, space-like or time-like according as the cosmological constant Λ was zero, positive or negative. While most applications of the idea of a conformal boundary have been to the zero- Λ , asymptotically Minkowskian case, there also has been work on the non-zero cases. Here, we review work with a positive Λ , which is the appropriate case for cosmology of the universe in which we live. This article is part of a discussion meeting issue ‘At the interface of asymptotics, conformal methods and analysis in general relativity’.