In the era of big data, rapid advancements in technology and data collection methods have led to the generation and accessibility of vast amounts of multi-modal, high-dimensional data across a diverse range of disciplines. Tensor methods have emerged as essential tools in signal processing, providing powerful frameworks to model and analyze such complex data effectively. This survey offers a comprehensive overview of tensor factorization techniques and their applications in key areas. We explore their role in remote sensing, focusing on tensor-based methods for analyzing hyperspectral and multispectral images, tackling challenges such as recovering super-resolution images and addressing spectral unmixing. In wireless communication, we examine tensor methods used for signal modulation in unsourced massive random access communication, which achieve strong performance in multi-antenna channel and signal modeling. We also discuss tensor applications in network compression, where they reduce the computational demands of deep neural networks, making them more feasible for edge devices. Additionally, we highlight the use of tensor methods in high-dimensional missing data completion problems, showcasing their versatility across various domains. Furthermore, we explore applications in image analysis and computer vision, where tensors are effectively utilized for motion and object tracking, 3D modeling, satellite image analysis, and medical imaging. By bridging theoretical advancements with practical applications, this survey aims to guide researchers in leveraging tensor methods to tackle emerging challenges in signal processing.