Spacetimes with negative cosmological constant are of interest both from a mathematical point of view, but also from a physical perspective in view of the conjectured AdS/CFT correspondence. A crucial feature of these spacetimes is their timelike null infinity, on which boundary conditions must be imposed. I will discuss several results in the theory of linear fields on anti-de Sitter backgrounds, including renormalisation and well-posedness, quasinormal modes and black hole stability.