The presence of multidirectional correlations in emerging multidimensional data poses a challenge to traditional regression modeling methods. Traditional modeling methods based on matrix or vector, for example, not only overlook the data’s multidimensional information and lower model performance, but also add additional computations and storage requirements. Driven by the recent advances in applied mathematics, tensor regression has been widely used and proven effective in many fields, such as sociology, climatology, geography, economics, computer vision, chemometrics, and neuroscience. Tensor regression can explore multidirectional relatedness, reduce the number of model parameters and improve model robustness and efficiency. It is timely and valuable to summarize the developments of tensor regression in recent years and discuss promising future directions, which will help accelerate the research process of tensor regression, broaden the research direction, and provide tutorials for researchers interested in high dimensional regression tasks. The fundamentals, motivations, popular algorithms, related applications, available datasets, and software resources for tensor regression are all covered in this monograph. The first part focuses on the key concepts for tensor regression, mainly analyzing existing tensor regression algorithms from the perspective of regression families. Meanwhile, the adopted low rank tensor representations and optimization frameworks are also summarized. In addition, several extensions in online learning and sketching are described. The second part covers related applications, widely used public datasets and software resources, as well as some real-world examples, such as multitask learning, spatiotemporal learning, human motion analysis, facial image analysis, neuroimaging analysis (disease diagnosis, neuron decoding, brain activation, and connectivity analysis) and chemometrics. This survey can be used as a basic reference in tensor-regression-related fields and assist readers in efficiently dealing with high dimensional regression tasks.