We shall outline two ways of introducing the modification of Einstein's relativistic symmetries of special relativity theory - the Poincaré symmetries. The most complete way of introducing the modifications is via the noncocommutative Hopf-algebraic structure describing quantum symmetries. Two types of quantum relativistic symmetries are described, one with constant commutator of quantum Minkowski space coordinates (θ_μν-deformation) and second with Lie-algebraic structure of quantum

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... introducing so-called κ-deformation. The third fundamental constant of Nature - fundamental mass κ or length λ - appears naturally in proposed quantum relativistic symmetry scheme. The deformed Minkowski space is described as the representation space (Hopf-module) of deformed Poincaré algebra. Some possible perspectives of quantum-deformed relativistic symmetries will be outlined.