Properties of (skew-symmetric) conformal Yano-Killing tensors are reviewed. The examples of CYK tensors in Minkowski, Kerr, de Sitter and anti-de Sitter spacetimes are discussed. Basic facts and definitions of the spin-2 field and conformal Yano-Killing tensors are introduced. Application of those two objects provides a precise definition of quasi-local gravitational charge. It leads to geometric definition of the asymptotic flat spacetime: `strong asymptotic flatness', which guarantees well defined total angular momentum. Conformal rescaling of conformal Yano-Killing tensors and relations between Yano and CYK tensors are discussed. Pullback of these objects to a submanifold is used to construct all solutions of a CYK equation in anti-de Sitter and de Sitter spacetimes. Properties of asymptotic conformal Yano-Killing tensors are examined for asymptotic anti-de Sitter spacetimes. Explicit asymptotic forms of them are derived. The results are used to construct asymptotic chargesin asymptotic AdS spacetime. Well known examples like Schwarzschild-AdS, Kerr-AdS and NUT-AdS are examined carefully in the construction of the concept of energy, angular momentum and dual mass in asymptotic AdS spacetime. Other applications: symmetric Killing tensors and constants of motion along geodesics.