A smooth Fano manifold admits a Kahler-Einstein metric if and only if is K-polystable (K-stable if the automorphism group of is finite). For smooth del Pezzo surfaces (two-dimensional Fano manifolds), Tian proved that the K-polystability is equivalent to the reductivity of the automorphism group. In this talk, I will explain which smooth Fano threefolds (three-dimensional Fano manifolds) are K-polystable and which are not K-polystable. This is a joint project with Carolina Araujo (IMPA), Ana-Maria Castravet (Versailles), Kento Fujita (Osaka), Anne-Sophie Kaloghiros (London), Jesus Martinez-Garcia (Essex), Constantin Shramov (Moscow), Hendrik Suess (Jena) and Nivedita Viswanathan (Loughborough).