One of the most important problems in the mathematical study of general relativity is the stability or instability properties of black holes. I will present recent work, obtained in collaboration with Gustav Holzegel (Imperial College London), concerning the decay property of linear fields outside a Kerr-Anti-de-Sitter black holes. Contrary to the asymptotically flat case, the fields only exhibit slow decay (logarithmic) in time. The talk will contain a general introduction to the subject.