In this article we review applications of the concepts of the Jahn-Teller effect in molecular magnetism. The scope of the contemporary field of molecular magnetism and its fascinating applications are shortly described. The theoretical background of molecular magnetism as well as the applications of molecular magnets are closely related to the basic concepts of the Jahn-Teller effect through their structural properties affecting magnetic anisotropy, interaction with light, photo-induced magnetism, co-operative behavior of molecule-based magnetic systems, and dynamical properties affected by relaxation processes and spin coherence times. We show that a wide class of symmetric spin-frustrated systems are orbitally degenerate, and the Jahn-Teller effect plays an important role in the description of their properties. In high-nuclearity magnetic clusters (single molecule magnets) the Jahn-Teller coupling stabilizes a specific alignment of the local magnetic axes, giving rise to a global anisotropy and consequently to a spin reorientation barrier. The problem of the double exchange in mixed-valence systems is considered, with the emphasis on the underlying role of the pseudo Jahn-Teller coupling in localization/delocalization of the mobile electron. Under certain conditions the latter gives rise to a reduction of the double exchange and, in particular, reduces the magnetic anisotropy in the presence of orbital degeneracy. The properties of mixed-valence systems are closely related to a complicated interplay between the pseudo Jahn-Teller interaction, isotropic exchange and double exchange. Manifestations of the Jahn-Teller effect are discussed for a wide class of photoactive (photoswitchable) systems. Pseudo Jahn-Teller models are employed for the description of the tautomeric transformations and extremely long living metastable states in photochromic compounds. Finally, we review the problem of co-operative phenomena in molecule-based extended mixed-valence systems, for which the Jahn-Teller mechanism is shown to result in the charge and structural ordering. The concept of the Jahn-Teller effect combined with the so-called quasidynamical approach allows to describe the intervalence optical bands and to reveal the underlying physical mechanism (quantum resonances of the vibronic levels) of the intricate quantum phenomena of the coexistence of localized and delocalized states in crystals based on interacting mixed-valence units.