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Under suitable experimental conditions collective spin-wave excitations,magnons, form a Bose-Einstein condensate (BEC) where the spins precess with aglobally coherent phase. Bose-Einstein condensation of magnons has beenreported in a few systems, including superfluid phases of $^3$He, solid statesystems such as Yttrium-iron-garnet (YIG) films, and cold atomic gases. Amongthese systems, the superfluid phases of $^3$He provide a nearly ideal testbench for coherent magnon physics owing to experimentally proven spinsuperfluidity, the long lifetime of the magnon condensate, and the versatilityof the accessible phenomena. We first briefly recap the properties of thedifferent magnon BEC systems, with focus on superfluid $^3$He. The main body ofthis review summarizes recent advances in application of magnon BEC as alaboratory to study basic physical phenomena connecting to diverse areas fromparticle physics and cosmology to new phases of condensed matter. This line ofresearch complements the ongoing efforts to utilize magnon BECs as probes andcomponents for potentially room-temperature quantum devices. In conclusion, weprovide a roadmap for future directions in the field of applications of magnonBEC to fundamental research.

Magnon Bose-Einstein condensates: from time crystals and quantum chromodynamics to vortex sensing and cosmology