The nickel-based 327 superconductor has emerged as a novel high-temperature superconducting system following cuprates and iron-based superconductors, attracting extensive research interest due to its superconducting critical temperature entering the liquid nitrogen range under high pressure. From the perspective of weak-coupling theory, this review systematically summarizes recent theoretical advances concerning the superconducting pairing mechanism and symmetry in nickel-based 327 superconductors. It highlights the application of methods including the random phase approximation, fluctuation exchange approximation, and functional renormalization group in analyzing the electronic structure, spin fluctuations, and effective pairing interactions in this system. Research indicates that within the bilayer two-orbital model, the Fermi surface nesting structure, particularly the γ pocket composed of d_3z²-r² orbitals, plays a crucial role in promoting spin fluctuations and superconducting pairing. Most weak-coupling theories predict the dominant superconducting pairing symmetry to be s_±-wave, characterized by a gap function with the same sign on the γ and α pockets but the opposite sign on the β pocket. Furthermore, studies have elucidated the effects of crystal field splitting, interlayer coupling, and pressure on the pairing symmetry, consistent with experimental observations. This review consolidates the achievements of weak-coupling theories in revealing the superconducting mechanism of nickel-based 327 superconductors and suggests that future work should integrate strong-coupling theories with more precise experimental data to further clarify the origin of superconductivity.