Spin Superfluidity and Magnone BEC in He-3*

The spin superfluidity -- superfluidity in the magnetic subsystem of a condensed matter -- is manifested as the spontaneous phase-coherent precession of spins first discovered in 1984 in $^3$He-B. This superfluid current of spins -- spin supercurrent -- is one more representative of superfluid currents known or discussed in other systems, such as the superfluid current of mass and atoms in superfluid $^4$He; superfluid current of electric charge in superconductors; superfluid current of hypercharge in Standard Model of particle physics; superfluid baryonic current and current of chiral charge in quark matter; etc. Spin superfluidity can be described in terms of the Bose condensation of spin waves -- magnons. We discuss different states of magnon superfluidity with different types of spin-orbit coupling: in bulk $^3$He-B; magnetically traped ``$Q$-balls'' at very low temperatures; in $^3$He-A and $^3$He-B immerged in deformed aerogel; etc. Some effects in normal $^3$He can also be treated as a magnetic BEC of fermi liquid. A very similar phenomena can be observed also in a magnetic systems with dinamical frequensy shift, like $MnC0_3$. We will discuss the main experimental signatures of magnons superfluidity: (i) spin supercurrent, which transports the magnetization on a macroscopic distance more than 1 cm long; (ii) spin current Josephson effect which shows interference between two condensates; (iii) spin current vortex -- a topological defect which is an analog of a quantized vortex in superfluids, of an Abrikosov vortex in superconductors, and cosmic strings in relativistic theories; (iv) Goldstone modes related to the broken $U(1)$ symmetry -- phonons in the spin-superfluid magnon gas; etc. For recent review see Yu. M. Bunkov and G. E. Volovik J. Phys. Cond. Matter. {\bf 22}, 164210 (2010)