Einstein's introduction of the cosmological constant to general relativity provided a mathematical framework for the study of the universe in the large. After a discussion of the early discoveries and ruminations of A Einstein and W de Sitter, I will move on to describe the Kerr de Sitter geometry which models a rotating black hole in an expanding universe. I will motivate the Cauchy problem in the context of the Einstein vacuum equations with positive cosmological constant, and present a recent resolution of the non-linear stability problem for the cosmological region. Among other works, the talk describes contributions by H Friedrich, P Hintz and A Vasy, and my recent joint work with G Fournodavlos.