Reduced-order models benefit the simulation, design, optimization, response prediction, and nonlinear system control of large flexible structures, and have attracted great attention in nonlinear dynamics research. Two critical issues arise in model reduction: it is needed to ensure high computational accuracy and efficiency, and more importantly, it is desired to be able to reproduce the nonlinear dynamic phenomena of the original full-order model, including instability, bifurcation, and complex dynamics such as quasi-periodic solutions, chaotic solutions, and wave turbulence. Normal form theory is a significant theory for the stability and bifurcation analysis of nonlinear dynamic systems, and its application in model reduction has seen substantial breakthroughs in recent years. This paper provides a brief introduction to normal form theory and reviews the progress in applying it for reduced-order modeling of geometric nonlinear systems. A special emphasis is put on the reduction method employing normal form theory for the finite element models: the direct normal form, and the direct parameterization of invariant manifolds. Additionally, the paper anticipates further advancements in these model reduction techniques.