We consider two parallel separately controlled graphene monolayers in which external gates induce a finite density of electrons in one layer and holes in another layer. In this thesis the theory of the excitonic insulator in such systems is developed. We analyze the symmetry of the excitonic state in the system, classify all possible phases, and build a phase diagram that takes into account the effect of the symmetry breaking due to the external electric and magnetic fields. The large-N approximation is used, where N = 8 is a number of electron's species in the system. Taking into account leading and sub-leading orders in 1/N expansion of the dynamically screened Coulomb interaction in the system, for the most energetically stable phase in absence of magnetic field we find a gap in the single-particle excitation spectrum at zero temperature. The result for the gap is found up to the first sub-leading order in 1/N. We determined the leading 1/N contribution to the exponential pre-factor in the expression for the gap Δ=Cexp[-2N]EF . Also we consider the simplified interaction, which reflects two most important features of the dynamically screened interaction: 1 )the interaction between charge carriers on Fermi surface, which is the most important for the excitonic condensation, and 2)the undamped plasmon. In such a case, in contrast to the suggested by other authors Ref.[59] enhancement of transition temperature Tc by plasmon pole in the dynamically screened interaction, we find no enhancement of Tc and arrive at Tc≈10-7EF, which is comparable to the result Tc≈10-7EF which was obtained using a statically screened interaction [42, 43]. Moreover, we found the presence of Goldstone modes with linear spectrum, therefore fluctuations of the order parameter suppress the transition into the excitonic insulator state even further.