Vinogradov first developed the method of trigonometric sums in the first decades of the twentieth century as a way of solving problems in analytical number theory. The authors here present a systematic account of the theory of multiple trigonometric sums, using a unified approach to derive results similar to those of Vinogradov, with the understanding the theory of multiple trigonometric sums has reached the level of completion of one-dimensional sums. They use their results to solve problems in analytic number theory and to investigate trigonometric integrals common in physics, statistics and analysis. They also present purely mathematical results for the solvability of equations in integers. This text is designed for graduate students and researchers in number theory, probability theory, and analysis

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