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This book is devoted to various applications of classical and functional analysis. In the first chapter moment problems and related problems are discussed. Connections to other fields (operator theory, Hahn - Banach type theorems, approximation, Krein - Milman theorem) are pointed out. The existence of a (the) solution is studied. In the end, some results on the uniqueness (or non - uniqueness) of the solution are recalled. Chapter II contains a few results on invariant subspaces and balls of bounded linear operators. Chapter III is devoted to elements of optimization theory. Chapter IV contains applications of Newton's method to approximation of the solutions of concrete scalar and operatorial equations. Most of the results belong to the authors of the book. This work addresses to researchers, professors, post - doc and students.