In condensed matter physics, the Fermi-Hubbard model is a fundamental model for describing interacting many-electron systems and is closely related to quantum many-body phenomena in correlated electron materials, particularly quantum magnetism and high-temperature superconductivity. Compared with the extensively studied two-dimensional systems, the three-dimensional (3D) Fermi-Hubbard model remains less explored. In recent years, however, with the successful experimental realization of its finite-temperature antiferromagnetic phase transition in ultracold atomic optical lattice, this 3D model has been gradually attracting growing interest. This article provides a concise and timely overview of the major theoretical progress on the three-dimensional Fermi-Hubbard model from research groups worldwide, analyzes its significance for understanding the fundamental physics of strongly correlated electron systems, and discusses several important open questions that deserve further attention.

The paper is organized as follows. We begin with a brief introduction of the Fermi-Hubbard model, emphasizing its fundamental role as a minimal paradigm for strongly correlated electron systems, and summarize its recent progress in both theoretical and experimental studies. Next, we provide a comprehensive review of the most recent determinantal quantum Monte Carlo (DQMC) studies of the half-filled 3D Hubbard model. This includes a detailed examination of the antiferromagnetic (AFM) and thermodynamic properties at finite temperature, with particular focus on the AFM phase transition and the metal-insulator crossover (MIC) phenomenon emerging above the Néel transition. The temperature dependences of commonly used observables are also discussed, including the double occupancy, specific heat, and charge susceptibility. In the third section, we review several many-body computational studies on the magnetic phase diagram of the 3D doped Hubbard model, including the AFM and spin-density wave phases. Finally, we highlight the outstanding challenges in the 3D doped Hubbard model, with an emphasis on the possible emergence of fermion pairing and unconventional superconductivity. We also discuss the direction of combining many-body precision computation with quantum simulation based on ultracold atomic optical lattice experiments to jointly address the major outstanding scientific changes in the 3D Fermi-Hubbard model.