Book excerpt: Transition metal oxides (TMOs) are an ideal arena for the study of electronic correlations because the s-electrons of the transition metal ions are removed and transferred to oxygen ions, and hence the strongly correlated d-electrons determine their physical properties such as electrical transport, magnetism, optical response, thermal conductivity, and superconductivity. These electron correlations prohibit the double occupancy of metal sites and induce a local entanglement of charge, spin, and orbital degrees of freedom. This gives rise to a variety of phenomena, e.g., Mott insulators, various charge/spin/orbital orderings, metal-insulator transitions, multiferroics, and superconductivity. In recent years, there has been a burst of activity to manipulate these phenomena, as well as create new ones, using oxide heterostructures. Most fundamental to understanding the physical properties of TMOs is the concept of symmetry of the order parameter. As Landau recognized, the essence of phase transitions is the change of the symmetry. For example, ferromagnetic ordering breaks the rotational symmetry in spin space, i.e., the ordered phase has lower symmetry than the Hamiltonian of the system. There are three most important symmetries to be considered here. (i) Spatial inversion (I), defined as r → -r. In the case of an insulator, breaking this symmetry can lead to spontaneous electric polarization, i.e. ferroelectricity, or pyroelectricity once the point group belongs to polar group symmetry. (ii) Time-reversal symmetry (T) defined as t → -t. In quantum mechanics, the time-evolution of the wave-function? is given by the phase factor e{sup -iEt/{h_bar}} with E being the energy, and hence time-reversal basically corresponds to taking the complex conjugate of the wave-function. Also the spin, which is induced by the 'spinning' of the particle, is reversed by time-reversal. Broken T-symmetry is most naturally associated with magnetism, since the spin operator changes sign with T-operation. (iii) Gauge symmetry (G), which is associated with a change in the phase of the wave-function as? → e{sup i{theta}}?. Gauge symmetry is connected to the law of charge conservation, and broken G-symmetry corresponds to superconductivity/superfluidity. To summarize, the interplay among these electronic degrees of freedom produces various forms of symmetry breaking patterns of I, T, and G, leading to novel emergent phenomena, which can appear only by the collective behavior of electrons and cannot be expected from individual electrons. Figure 1 shows this schematically by means of several representative phenomena. From this viewpoint, the interfaces of TMOs offer a unique and important laboratory because I is already broken by the structure itself, and the detailed form of broken I-symmetry can often be designed. Also, two-dimensionality usually enhances the effects of electron correlations by reducing their kinetic energy. These two features of oxide interfaces produce many novel effects and functions that cannot be attained in bulk form. Given that the electromagnetic responses are a major source of the physical properties of solids, and new gauge structures often appear in correlated electronic systems, we put 'emergent electromagnetism' at the center of Fig. 1.