This book consists of three parts. In the first the author outlines some classical and contemporary models of probability theory (such as those of Bernoulli, Gauss, Poisson, Renyi, de Finetti, etc.) pointing out their common traits and flaws (from the author's point of view). In the second part a probabilistic theory for a system of ordered events is given. The third part of the book deals with a new theory initiated by the author. This theory is constructed directly on the set of events and its elements are, instead of random variables as in the classical theory, the integrals taken over the set of events and called sum-functions. Some aspect of probability such as convergence in various senses, the central limit problem, martingales, etc., are treated in accordance with this new theory. At the end, a paper of I Cuculescu on supermartingales from the point of view of sum-functions, is added to the book.