This text introduces at a moderate speed and in a thorough way the basic concepts of the theory of stochastic integrals and Ito calculus for sem i- martingales. There are many reasons to study this subject. We are fascinated by the contrast between general measure theoretic arguments and concrete probabilistic problems, and by the own flavour of a new differential calculus. For the beginner, a lot of work is necessary to go through this text in detail. As areward it should enable her or hirn to study more advanced literature and to become at ease with a couple of seemingly frightening concepts. Already in this introduction, many enjoyable and useful facets of stochastic analysis show up. We start out having a glance at several elementary predecessors of the stochastic integral and sketching some ideas behind the abstract theory of semimartingale integration. Having introduced martingales and local martingales in chapters 2 - 4, the stochastic integral is defined for locally uniform limits of elementary processes in chapter S. This corresponds to the Riemann integral in one-dimensional analysis and it suffices for the study of Brownian motion and diffusion processes in the later chapters 9 and 12.