The excitonic insulator (EI) is a new state matter caused by the exciton condensation, which has been recently discovered in some two dimensional hetero-structures. In this lecture, I will introduce two unique properties of EI, which make it fundamentally different with ordinary band insulators. The first property of EI is a new electrical breakdown mechanism for EIs in the BCS limit, which differs fundamentally from the Zener breakdown mechanism observed in traditional band insulators. Our new mechanism results from the instability of the many-body ground state for exciton condensation, caused by the strong competition between the polarization and condensation energies in the presence of an electric field. We refer to this mechanism as “many-body breakdown”. The second properties of EI is the unique behavior of quantum capacitance. Our theoretical study reveals that after exciton condensation the capacitance will be strongly modified by the Fock process of the Coulomb interaction and deviates greatly from its classical (or geometry) value. Moreover, with the presence of magnetic field such a quantum capacitance oscillates strongly with the inverse of field strength 1/B even in an insulator phase, which can be viewed as the “smoking gun” evidence for a EI.