Sums and Gaussian Vectors Surveys the methods currently applied to study sums of infinite dimensional independent random vectors in situations where their distributions resemble Gaussian laws. Covers probabilities of large deviations Chebyshev type inequalities for seminorms of sums a method of constructing Edgeworth type expansions estimates of characteristic functions for random vectors obtained by smooth mappings of infinite dimensional sums to Euclidean spaces. A self contained exposition of the modern research apparatus around CLT the book is accessible to new graduate students and can be a useful reference for researchers and teachers of the subject