A superconductor is a material that conducts electric current with no resistance. Superconductivity and magnetism are known to be antagonistic phenomena: superconductors expel weak external magnetic field (the Meissner effect) while a sufficiently strong magnetic field, in general, destroys superconductivity. In a seemingly contradictory statement, we show that a very strong magnetic field can turn an empty space into a superconductor. The external magnetic field required for this effect should be about 10^{16} Tesla (eB ~ 1 GeV^2). The physical mechanism of the exotic vacuum superconductivity is as follows: in strong magnetic field the dynamics of virtual quarks and antiquarks is effectively one-dimensional because these electrically charged particles tend to move along the lines of the magnetic field. In one spatial dimension a gluon-mediated attraction between a quark and an antiquark of different flavors inevitably leads to formation of a colorless spin-triplet bound state (a vector analogue of the Cooper pair) with quantum numbers of an electrically charged rho meson. Such quark-antiquark pairs condense to form an anisotropic inhomogeneous superconducting state similar to the Abrikosov vortex lattice in a type-II superconductor. The onset of the superconductivity of the charged rho mesons should also induce an inhomogeneous superfluidity of the neutral rho mesons. The vacuum superconductivity should survive at very high temperatures of typical Quantum Chromodynamics (QCD) scale of 10^{12} K (T ~ 100 MeV). We propose the phase diagram of QCD in the plane "magnetic field - temperature".