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Color superconductivity is expected to be formed in high density quark matterwhere color symmetry is spontaneously broken in the presence of di-quarkcondensate. Stable non-Abelian vortices or color magnetic flux tubes exist inthe color-flavor locked phase at asymptotically high density. $\mathbb{C}P^2$Nambu-Goldstone(NG) bosons and Majorana fermions belonging to the tripletrepresentation are localized around a non-Abelian vortex. We discuss the zeromode analysis and the low-energy effective world sheet theory of a non-Abelianvortex. We determine the interactions of these bosonic and fermionic modes byusing the nonlinear realization method. We also discuss the Aharanov-Bohm (AB)phases of charged particles, such as, electrons, muons, and color-flavor lockedmesons made of tetra-quarks encircling around a non-Abelian vortex in thepresence of electro-magnetic fields. This is a review based on our recent worksarXiv:1602.01677, arXiv:1512.06603, arXiv:1612.09419.

Stable non-Abelian semi-superfluid vortices in dense QCD