The classical concept for emergence of Turing patterns in reaction-diffusion systems requires that a system should be composed of complementary subsystems, one of which is unstable and diffuses sufficiently slowly while the other one is stable and diffuses sufficiently rapidly. In this work, the phenomena of emergence of Turing patterns are studied and do not fit into this concept, yielding the following results. (1) The criteria are derived, under which a reaction-diffusion system with immobile species should spontaneously produce Turing patterns under any diffusion coefficients of its mobile species. It is shown for such systems that under certain sets of types of interactions between their species, Turing patterns should be produced under any parameter values, at least provided that the corresponding spatially non-distributed system is stable. (2) It is demonstrated that in a reaction-diffusion system, which contains more than two species and is stable in absence of diffusion, the presence of a sufficiently slowly diffusing unstable subsystem is already sufficient for diffusion instability (i.e., Turing or wave instability), while its complementary subsystem can also be unstable. (3) It is shown that the presence of an immobile unstable subsystem, which leads to destabilization of waves within an infinite range of wavenumbers, in a spatially discrete case can result in the generation of large-scale stationary or oscillatory patterns. (4) It is demonstrated that under the presence of subcritical Turing and supercritical wave bifurcations, the interaction of two diffusion instabilities can result in the spontaneous formation of Turing structures outside the region of Turing instability.