This book mainly deals with the BochnerOCoRiesz means of multiple Fourier integral and series on Euclidean spaces. It aims to give a systematical introduction to the fundamental theories of the BochnerOCoRiesz means and important achievements attained in the last 50 years. For the BochnerOCoRiesz means of multiple Fourier integral, it includes the Fefferman theorem which negates the Disc multiplier conjecture, the famous Carleson-SjAlin theorem, and Carbery-Rubio de Francia-Vega's work on almost everywhere convergence of the BochnerOCoRiesz means below the critical index. For the BochnerOCoRiesz means of multiple Fourier series, it includes the theory and application of a class of function space generated by blocks, which is closely related to almost everywhere convergence of the BochnerOCoRiesz means. In addition, the book also introduce some research results on approximation of functions by the BochnerOCoRiesz means.