Many modern statistical and machine learning problems involve parameters that are intrinsically multiway, including spatiotemporal arrays, network data, and weight tensors in large neural networks. Although such objects can always be vectorized, doing so ignores multilinear structure and can lead to substantially different statistical and computational behavior. Weight tensors from large language models will serve as a motivating example, illustrating how tensor methods can preserve structural information that is lost under naive vectorization. This talk develops a unified perspective on recent work in tensor-structured learning through three themes: compression, communication, and completion. Methodologically, I will discuss low-rank tensor parameterizations, rank selection through optimism-based risk analysis, communication-efficient randomized tensor algorithms, and structured sampling schemes for tensor completion. A recurring goal is to obtain nonasymptotic guarantees that connect tensor geometry, statistical error, computational cost, and sample or communication complexity with formal guarantees.