The properties of nuclear matter at zero and finite temperatures in the frame of the Brueckner theory realistic nucleon-nucleon potentials are studied. Comparison with other calculations is made. In addition we present results for the symmetry energy obtained with different potentials, which is of great importance in astrophysical calculation. Properties of asymmetric nuclear matter are derived from various many-body approaches. This includes phenomenological ones like the Skyrme Hartree-Fock and relativistic mean field approaches, which are adjusted to fit properties of nuclei, as well as more microscopic attempts like the BHF approximation, a Self-Consistent Greens Function (SCGF) method and the so-called Vlowk approach, which are based on realistic nucleon-nucleon interactions which reproduce the nucleon-nucleon phase shifts. These microscopic approaches are supplemented by a density-dependent contact interaction to achieve the empirical saturation property of symmetric nuclear matter. Special attention is paid to behavior of the isovector and the isoscalar component of the effective mass in neutron-rich matter. The nuclear symmetry potential at fixed nuclear density is also calculated and its value decreases with increasing the nucleon energy. In particular, the nuclear symmetry potential at saturation density changes from positive to negative values at nucleon kinetic energy of about 200 MeV. The hot properties of nuclear matter are also calculated using T2–approximation method at low temperatures. Good agreement is obtained in comparison with previous theoretical estimates and experimental data especially at low densities.