Attractor is an attracting set of a dynamical system. This tautological definition requires formalization. There are different non-equivalent definitions of the attractors of diffeomorphisms: maximal attractors (Auslender et al, 1964), Milnor attractor (1985), statistical and minimal attractors (the speaker, 1986 - 96). Maximal attractors are Lyapunov stable and topologically defined. Milnor and other attractors are not, even for generic dynamical systems (Shilin, 2017; Bonatti, Minkov, Okunev, Shilin, in preparation). These results will be delivered, together with the definitions of the attractors named above.