The subject of this thesis is the historical development of the theory of complex numbers and functions, from the first use of imaginary numbers up to the establishment of complex analysis. Complex numbers first appeared in the Renaissance period, when mathematicians Cardano and Bombelli, solving cubic equations, realized that, in order to get real solutions it was sometimes necessary to use square roots of negative numbers, and described how to calculate with them. This is the topic of the first chapter of this thesis. The second chapter describes the progress made during the 17th and 18th century when mathematicians tried to find their geometric interpretation, which was finally successfully done by Wessel, Gauß and Argandat the turn of the 18th to the 19th century. This period brought significant results for calculation with complex numbers, e.g. the Euler product and the De Moivre formula. The last, third, chapter describes how Cauchy, Laurent, Riemann and Weierstraß established the theory complex functions, i.e. complex analysis.