In this talk we discuss our recent suggestion that the QCD vacuum in a sufficiently strong magnetic field (stronger than 10^{16} Tesla) may undergo a spontaneous transition to an electromagnetically superconducting state. The possible superconducting state is anisotropic (the vacuum exhibits superconductivity only along the axis of the uniform magnetic field) and inhomogeneous (in the transverse directions the vacuum structure shares similarity with the Abrikosov lattice of an ordinary type-II superconductor). The electromagnetic superconductivity of the QCD vacuum is suggested to occur due to emergence of specific quark-antiquark condensates which carry quantum numbers of electrically charged rho mesons. A Lorentz-covariant generalization of the London transport equations for the magnetic-field-induced superconductivity is given.