The book presents an in—depth study of arbitrary one—dimensional continuous strong Markov processes using methods of shastic calculus. Departing from the classical approaches, a unified investigation of regular as well as arbitrary non—regular diffusions is provided. A general construction method for such processes, based on a generalization of the concept of a perfect additive functional, is developed. The intrinsic decomposition of a continuous strong Markov semimartingale is discovered. The book also investigates relations to shastic differential equations and fundamental examples of irregular diffusions.