The physics program and the present status of the project of NIC A accelerator complex, which is under construction at JINR (Dubna), are presented. The main goal of the project is to provide ion beams for experimental studies of hot and dense strongly interacting baryonic matter and spin physics. The proposed physics program concentrates on the search for possible manifestations of the phase transitions and critical phenomena in the energy region, where the excited matter is produced with maximal achievable net baryon density, and clarification of the origin of nucleon spin. The NICA collider will provide heavy ion collisions in the energy range of √/s_NN=4÷11 GeV at average luminosity of L=110²⁷cm²s^-1 for ¹⁹⁷Au⁷⁹⁺ nuclei and polarized proton collisions in energy range of √s_nn=12÷27 GeV at luminosity of L≥ 10³²cm^2-s^-1. The key issue of the accelerator complex is application of sophisticated beam accumulation schemes and both stochastic and electron cooling methods. Strong space-charge effects in the collider arise a challenge to its optics and application of novel methods of beam stability maintenance.