The European Spallation Neutron Source (ESS) delivers high-intensity pulsed particle beams with 5-MW average beam power at 1.3-GeV incident proton energy. This causes sophisticated demands on material and geometry choices and a very careful optimization of the whole target system. Therefore, complex and detailed particle transport models and computer code systems have been developed and used to study the nuclear assessment of the ESS target system. The purpose here is to describe the methods of calculation mainly based on the Monte Carlo code to show the performance of the ESS target station. The interesting results of the simulations of the mercury target system are as follows: time-dependent neutron flux densities, energy deposition and heating, radioactivity and afterheat, materials damage by radiation, and high-energy source shielding. The results are discussed in great detail. The validity of codes and models, further requirements to improve the methods of calculation, and the status of running and planned experiments are given also.