The Nuclotron-based Ion Collider fAcility (NICA) is under construction at JINR. The NICA goal is to provide of colliding beams for studies of hot and dense strongly interacting baryonic matter and spin physics. The ion mode accelerator facility of the NICA Collider consists of the following accelerators: The new operating Heavy Ion Linac (HILAC) with RFQ and IH DTL sections at energy 3.2 MeV/u, new operating superconducting Booster synchrotron at energy up 600 MeV/u, operating superconducting synchrotron Nuclotron for the gold ion energy 3.9 GeV/u and two Collider storage rings with two interaction points. There is the electron cooling system in the Booster synchrotron, the Collider has electron and stochastic cooling systems. The status of the NICA acceleration complex and its cooling systems is presented. The application of the cooling systems to the operation of the NICA accelerators - the Booster and the Nuclotron are discussed.