In data analysis, the covariance matrix plays a pivotal role in studying the relationships between a set of variables. It has been commonly and widely used across various fields regardless of the data being dense or sparse, low-dimension or high-dimension, discrete or continuous, fixed or random, time series or non-time series, structured or non-structured, and so forth. Hence, the main theme of this book is “covariance.” To study this subject, we first introduce multivariate data and its expressions, along with two examples. We next state the scope and the contributions of the book and then briefly describe each of the next nine chapters’ key points. Finally, we present some notation, norms, operations, decompositions, order operations, asymptotic properties, and inequalities that will be directly and indirectly used in this book and its associated references.