Omics technologies, including genomics, transcriptomics, proteomics, and metabolomics, have revolutionized biological research by enabling comprehensive, high-throughput analysis of molecular components within cells and organisms. The resulting high-dimensional datasets pose significant analytical challenges, particularly in integrating diverse data types and uncovering complex biological relationships. Tensor-based approaches have emerged as powerful tools for analyzing these high-dimensional omics datasets, offering advantages over traditional matrix-based methods in capturing complex, multi-way relationships. This review provides an overview of tensor decomposition techniques and their applications in omics data analysis, with a focus on multi-sample gene expression data, multi-omics integration, data imputation, and inference of cell-cell interactions from single-cell RNA sequencing. We discuss how tensors can naturally represent multidimensional omics datasets and how tensor factorization methods enable dimensionality reduction while preserving important structural information. A comprehensive biological background and an overview of relevant public databases and resources are provided to contextualize the computational methods. Case studies are presented to illustrate the application of tensor methods for tasks such as identifying gene expression modules, integrating multiple types of omics data, imputing missing values, and uncovering ligand-receptor interaction patterns. We highlight how tensor approaches can reveal higher-order interactions and context-dependent relationships that may be missed by traditional analyses. Challenges and future directions for tensor-based omics data analysis are also discussed, emphasizing the potential of these methods to extract meaningful biological insights from complex, heterogeneous datasets and advance our understanding of biological systems.