of the BEC, most of which comprise Cu dimers.

The focus of this article is to review systematic concepts and experimental realizations of a transition to a BEC phase in DTN, where bosons are quanta of magnetic excitations in a magnetically ordered ensemble of magnetic moments. Although there is no consensus on a specific name for the magnetic excitations, we will use the word magnon for the excitations in insulating systems generated by the application of a magnetic field. This denomination follows the nomenclature used by the first papers on BEC in magnetic insulators, since elementary excitations in antiferromagnetic systems are called magnons [@2; @7].

We hope that the experimental evidence for BEC in DTN will stimulate new research in this fascinating quantum phenomenon.

The outline of this paper is as follows. Section 2 reviews the concepts of the ideal BEC at finite temperature and the nature of BEC when magnons are induced by the application of a magnetic field. Section 3 presents the magnetic properties of DTN, introducing the Hartree-Fock-Popov analysis of the hard-core boson Hamiltonian. Section 4 presents experimental works that lead to the interpretation of the induced phase diagram as a BEC. Section 5 summarizes the results 