Recently rapid progress has been achieved from several fronts in computational approaches for treating many-fermion systems.  With the exciting experimental developments from ultracold atoms, this brings new synergy for advancing our understanding of correlated fermions. I will describe work within one of the computational frameworks, the auxiliary-field quantum Monte Carlo (AFQMC), to determine the properties of two fundamental two-dimensional systems, the Fermi gas and the repulsive Hubbard model. In the Fermi gas, where the interaction is attractive, exact computations are performed under experimental conditions with ultracold atoms, and the interplay between superfluidity and Rashba spin-orbit coupling is investigated. In the repulsive Hubbard model, the magnetic and charge orders upon doping are examined and several key questions about the ground state are addressed.