The book is devoted to the foundations of the theory and methods for calculating Lebesgue integrals in linear topological spaces. A detailed proof of the convergence of sums of independent measurable maps in separable Fréchet spaces is given. Integrals on Gaussian and non-Gaussian measures are considered. The main facts from the theory of Hilbert spaces generated by Gaussian measures in linear topological spaces are given. Approximate formulas of various types are given for calculating functional integrals. Methods of approximate calculation of multiple functional integrals on the product of spaces are considered. Approximate formulas for calculating integrals by non-Gaussian measures are obtained, as well as formulas for integrals by operator-valued measures. Special methods for calculating functional integrals based on the use of interpolation functional polynomials, Feynman diagrams and interpolation of measures are considered. Some applications of functional integrals in quantum mechanics, quantum field theory, in nuclear physics, for the study of open quantum systems, the study of tunneling issues and topological effects are considered. The book is designed for mathematicians, physicists and other specialists, as well as graduate students and students of physical and mathematical specialties.