Nuclear matter equation of state (EoS) plays an important role in both nuclear physics and astrophysics. The first part of the present dissertation covers a microscopic derivation of a reliable nuclear matter EoS and its applications. To obtain a reliable nuclear matter EoS, we have employed in our derivations either a Brown-Rho (BR) scaled low-momentum nucleon-nucleon (NN) interaction V<sub>low-k</sub> or a combined interaction given by the sum of an unscaled V<sub>low-k</sub> and three-body forces. An all-order ring-diagram summation method is used in calculating the EoS so as to include the effect of the ground-state correlations, noting that such correlations are not included if the Hartree-Fock mean field method is employed. Neutron stars are a nature's unique laboratory for testing nuclear matter EoS. Their properties such as masses, radii and moments of inertia, can be calculated by solving the Tolman-Oppenheimer-Volkov equations using the nuclear matter EoS. We have tested our EoS in such calculations with and without including the effects from BR scalings. The neutron star masses, radii and moments of inertia given by the BR scaled EoS are in good agreement with empirical values, while those without such scalings are not acceptable. The <super>1</super>S<sub>0</sub> scattering length a<sub>s</sub> of the neutron-neutron interaction is -18.97 fm<super>-1</super>, which is quite large. What will happen when a<sub>s</sub> goes to infinity, reaching the so called &ldquo unitary limit &rdquo? We have studied the pure neutron matter's EoS with a family of unitarity potentials all of which are constructed to have infinite <super>1</super>S<sub>0</sub> scattering lengths. For such systems, a quantity of much interest is the universal ratio &xi = E<sub>0</sub>/(E<sub>0</sub>)<super>free</super> (where E<sub>0</sub> is the true ground-state energy of the system, and (E<sub>0</sub>)<super>free</super> is that for the non-interacting system). For all the unitarity potentials considered, we have numerically obtained &xi approx 0.44 for the cold neutron matter, supporting the existence of such a universality. The nuclear symmetry energy E<sub>sym</sub> is an important quantity which can also be derived from the nuclear matter EoS. We have calculated E<sub>sym</sub> up to densities of 4 ~ 5 n<sub>0</sub> with the effects from the BR and Brown-Rho-Ericson scalings included (n<sub>0</sub> represents the saturation density of symmetric nuclear matter). Our results indicate that E<sub>sym</sub> is monotonically increasing with the nuclear density. We have compared our results with the constraints on E<sub>sym</sub> deduced from heavy-ion collision experiments. The last part of this dissertation describes a new method for the derivation of the shell-model effective NN interactions. We have performed calculations of the sd and sdpf shell-model effective interactions using the Krenciglowa-Kuo and the recently developed extended Krenciglowa-Kuo iteration methods. We have also studied the effects of three-body forces on such interactions using a density-dependent two-body interaction derived from the chiral leading-order N<super>2</super>LO three-body forces.