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This article provides a discussion on the construction of conformal Gaussiangauge systems to study the evolution of solutions to the Einstein fieldequations with positive Cosmological constant. This is done by means of a gaugebased on the properties of conformal geodesics. The use of this gauge, combinedwith the extended conformal Einstein field equations, yields evolutionequations in the form of a symmetric hyperbolic system for which standardCauchy stability results can be employed. This strategy is used to study theglobal properties of de Sitter-like spacetimes with constant negative scalarcurvature. It is then adapted to study the evolution of the Schwarzschild-deSitter spacetime in the static region near the conformal boundary. This reviewis based on Class. Quantum Grav. 38 145026 and Class. Quantum Grav. 40 145005.

Conformal geodesics and the evolution of spacetimes with positive Cosmological constant