Textbook in PDF format Tensors are essential in modern day computational and Data Science. This book explores the foundations of tensor decompositions, a data analysis methodology that is ubiquitous in Machine Learning, signal processing, chemometrics, neuroscience, Quantum Computing, financial analysis, social science, business market analysis, image processing, and much more. In this self-contained mathematical, algorithmic, and computational treatment of tensor decomposition, the book emphasizes examples using real-world downloadable open-source datasets to ground the abstract concepts. Methodologies for 3-way tensors (the simplest notation) are presented before generalizing to d-way tensors (the most general but complex notation), making the book accessible to advanced undergraduate and graduate students in mathematics, Computer Science, statistics, engineering, and physical and life sciences. Additionally, extensive background materials in linear algebra, optimization, probability, and statistics are included as appendices. At its heart, this is a book about algorithms for tensor decompositions, helping readers to understand the most studied and used methods and trade-offs among them. Understanding algorithms requires understanding the theoretical nature of tensor decompositions. Certain tensor problems are known to be computationally difficult, but there are strategies for addressing many of the challenges. In the case of Tucker decomposition, for example, we show that some of the methods are quasi-optimal. This book is organized into four main parts. Part I (Tensor Basics) is introductory. The discussion of Tucker and CP (Parts II and III) are independent, so a course can focus on solely one or the other. Part IV (Closing Observations) is primarily for perspective and is entirely optional. We do not prescribe a specific computational platform, but everything described here can be computed using the Tensor Toolbox for MatLAB. Much of the same functionality is available in its Python clone, the Python Tensor Toolbox (PyTTB)