The properties of transition metal compounds are largely determined by the nontrivial interplay of different degrees of freedom: charge, spin, lattice, and also orbital ones. Especially rich and interesting effects occur in systems with orbital degeneracy. They result in the famous Jahn-Teller effect, leading to a plethora of consequences, in static and in dynamic properties, including nontrivial quantum effects. In the present talk, we will discuss the main phenomena in the physics of such systems, with particular attention to their novel manifestations. We will first briefly discuss the basic phenomena and their description, and then focus on several specific directions in this field. One of them is the reduction of effective dimensionality in many systems with orbital degrees of freedom due to the directional character of orbitals, with concomitant occurrence of some instabilities, leading, in particular, to the formation of dimers, trimers and similar clusters in the material. The properties of such cluster systems, largely determined by their orbital structure, are discussed in detail, and many specific examples of those in different materials are presented. Another large area that has gained particular importance relatively recently is the role of relativistic spin-orbit interaction. The mutual influence of this interaction and the more traditional Jahn-Teller physics are considered in detail in the second part of the review. In discussing all these issues, special attention is paid to the new quantum effects in them.

The research was supported by the Russian Science Foundation (project № 23-12-00159).

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