ABSTRACTNuclear norm, also known as trace norm, has been widely used in statistical machine learning. Nuclear norm regularization has emerged as an important tool for addressing various statistical problems involving the estimation of low‐rank matrices, particularly in tasks such as matrix completion and reduced rank regression. This review delves into the foundational models, practical implementations, and recent advancements in nuclear norm regularization. We discuss key implementation techniques, including semidefinite programming and singular value thresholding, which enable efficient solutions to low‐rank matrix estimation problems. Additionally, we examine the application of nuclear norm regularization in matrix covariate and matrix response regression, as well as its extension to tensor regression problems. Our study highlights the versatility and efficacy of nuclear norm regularization in providing both theoretical guarantees and scalable computational methods. Future research directions include improving computational efficiency, refining conditions for theoretical guarantees and extending applications to higher‐order tensors.