  The peculiarities in the Bose-Einstein condensation of particles and
quasiparticles are discussed. The difference between the condensation of
conserved and unconserved particles is analyzed. A classification of
quasiparticles is given. The emphasis is made on the ability of particles and
quasiparticles to condense. Illustrations include: general Bose-condensed
atomic systems, such as ensembles of trapped atoms, Bose gases with conserved
and unconserved number of atoms, vibrating atoms in double-well lattices,
Holstein-Primakoff magnons, Schwinger bosons, slave bosons, and the
condensation of singletons and triplons. The basic difference is that the
system of particles, whose total number is conserved, can form equilibrium as
well as nonequilibrium condensates, while unconserved particles can condense
only in a nonequilibrium system subject to external pumping supporting the
density of these particles sufficient for their condensation. The examples of
such a nonequilibrium condensation of unconserved particles are the
Bose-Einstein condensation of excitons, polaritons, and photons. Elementary
collective excitations, such as bogolons and phonons, being self-consistently
defined, do not condense. Magnons cannot condense in equilibrium systems.
Controversies, existing in literature with regard to the Bose-Einstein
condensation of some quasiparticles, are explained. Pushing a system out of
equilibrium may favor the condensation of unconserved quasiparticles, but
suppresses the condensate fraction of conserved particles.
