In this paper we review our recent study on muonium (Mu) diffusion in
simplest solids - Van der Waals cryocrystals. We give experimental evi
dence for the quantum nature of Mu diffusion in these matters. The res
ults are compared with the current theory of quantum diffusion in insu
lators. The first direct observation of the T7 power-law for the Mu ho
p rate (in solid nitrogen) is taken as confirmation of two-phonon scat
tering mechanism. In solid xenon the one-phonon interaction is shown t
o be dominant. Muonium diffusion in solid neon is discussed in terms o
f band-like motion. Finally, we report a dramatic effect of impurities
on Mu quantum diffusion.