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The main purpose of the present text is to show how mathematical analysis describes periodic functions and distributions. Along the five chapters the basis of trigonometry (in the real and complex numbers setting), of trigonometric polynomials, and of Fourier - trigonometric series are discussed. Simple periodic as well as double periodic (elliptic) functions are studied. One uses both classical and functional analysis, the latter being applied in the theory of trigonometric series, as well as in studding periodic distributions (which are linear continuous functionals on certain test - function spaces). The book may be useful to any reader interested in periodic problems. Minimal knowledge in real and complex mathematical analysis is necessary for reading this work.