This chapter presents the core method around which all the others will be organized, as well as its main variations: Principal Component Analysis (PCA). This method is the backbone of the book. The three main domains in which it can be developed are presented: algebra, with the approximation of a given matrix by a low rank matrix; Euclidean geometry, which is more intuitive and involves projecting a point cloud on a low-dimensional affine space; and statistics, where it involves exhibiting a correlation structure between variables. We recall that performing a PCA on a matrix A is equivalent to calculating its Singular Value Decomposition (SVD). This allows us to inherit all the advances made in the calculation of an SVD in high dimension, in particular by random projection as presented here, in the calculation of a PCA. We also introduce the main concepts (principal axis and principal coordinate) and the classical evaluations of the quality of the approximation which will be used throughout the book, i.e., by the variety of methods presented in the introduction.