The review discusses the theoretical aspects of ultracold atomic Fermi gases, focusing on their behavior in both uniform and harmonically trapped configurations. It emphasizes the role of interactions, which are characterized by the s-wave scattering length, and how they lead to superfluidity at low temperatures. Key regimes include the BCS limit, Bose-Einstein condensation of dimers, and the unitary limit of large scattering length. The review covers various physical properties such as density profiles, energy, momentum distribution, collective oscillations, and superfluidity signatures like quantized vortices. It also discusses the effects of interactions in different environments, including optical lattices and spin-polarized systems. Theoretical approaches range from mean-field descriptions to quantum Monte Carlo simulations. The review compares theoretical predictions with experimental results, highlighting the importance of the scattering length in tuning interactions and the universality of the unitary regime. It also addresses the dynamics of superfluidity, including expansion, collective oscillations, and the role of spin polarization. The review concludes with a discussion of the implications of these findings for future research in ultracold Fermi gases.The review discusses the theoretical aspects of ultracold atomic Fermi gases, focusing on their behavior in both uniform and harmonically trapped configurations. It emphasizes the role of interactions, which are characterized by the s-wave scattering length, and how they lead to superfluidity at low temperatures. Key regimes include the BCS limit, Bose-Einstein condensation of dimers, and the unitary limit of large scattering length. The review covers various physical properties such as density profiles, energy, momentum distribution, collective oscillations, and superfluidity signatures like quantized vortices. It also discusses the effects of interactions in different environments, including optical lattices and spin-polarized systems. Theoretical approaches range from mean-field descriptions to quantum Monte Carlo simulations. The review compares theoretical predictions with experimental results, highlighting the importance of the scattering length in tuning interactions and the universality of the unitary regime. It also addresses the dynamics of superfluidity, including expansion, collective oscillations, and the role of spin polarization. The review concludes with a discussion of the implications of these findings for future research in ultracold Fermi gases.