We study the non-Abelian topological vortices in condensed matter physics,
whose topological flux quantum number is described by $\pi_2(S^2)$, not by
$\pi_1(S^1)$. We present two examples, a magnetic vortex in two-gap
superconductor and a vorticity vortex in two-component Bose-Einstein
condensate. In both cases the condensates exhibit a global SU(2) symmetry which
allows the non-Abelian topology. We establish the non-Abelian flux quantization
in two-gap superconductor by demonstrating the existence of non-Abelian
magnetic vortex whose flux is quantized in the unit $4\pi/g$, not $2\pi/g$. We
also discuss a genuine non-Abelian gauge theory of superconductivity which has
a local SU(2) gauge symmetry, and establish the non-Abelian Meissner effect in
the non-Abelian superconductor. We compare the non-Abelian vortices with the
well-known Abelian Abrikosov vortex, and discuss how these non-Abelian vortices
could be observed experimentally in two-gap superconductor made of ${\rm
MgB_2}$ and spin-1/2 condensate of $^{87}{\rm Rb}$ atoms. Finally, we argue
that the existence of the non-Abelian vortices provides a strong evidence for
the existence of topological knots in these condensed matters whose topology is
fixed by $\pi_3(S^2)$, which one can construct by twisting and connecting the
periodic ends of the non-Abelian vortices.