I will give a gentle introduction with several examples to the following topic: Yau-Tian-Donaldson conjecture states that a polarised manifold $(X,L)$ admits a cscK metric in $c_1(L)$ if and only if $(X,L)$ is K-polystable. Matsushima proved in 1957 that existence of such cscK metric implies reductively of the automorphism group of $X$. In a positive direction on the YTD conjecture, we show that K-polystability implies reductively of the automorphism group of $X$, for smooth Fano threefolds. This is joint work with Paolo Cascini and Ivan Cheltsov.