This volume is a revised and enlarged version of Chapters 1 and 2 of a book with the same title, published in Romanian in 1968. The revision resulted in a new book which has been divided into two parts because of the large amount of new material. The present part is intended to introduce mathematicians and biologists with a strong mathematical and probabilistic background to the study of stochastic processes. We hope some readers will be able to discover by themselves the new features of our treatment such as the inclusion of some unusual topics, the special attention paid to some usual topics, and the grouping of the material. We draw the reader's attention to the numbering, because there are structural differences between the two parts. In Part I there are Chapters, Sections, Subsections, Paragraphs and Subparagraphs. Thus the numbering a. b. c. d. e refers to Subparagraph e of Paragraph d of Subsection c of Section b of Chapter a. Definitions, theorems lemmas and propositions are numbered a. b. n, n = 1,2, . . . , where a indicates the chapter and b the section. In Part II there are Sections, Subsections, Paragraphs, and SUbparagraphs. Thus the numbering a. b. c. d refers to Subparagraph d of Paragraph c of Subsection b of Section a. Theorems and lemmas are numbered a. n, n = 1, 2, . . . , where a indicates the section.