Atya-Singer's index theorem is one of the most popular mathematical results of the last five years. Such interest in the index problem is due to its position at the junction of analysis and topology, as well as the fact that it required the latest mathematical developments to solve it. Atya-Singer's Theorem and related issues are devoted to a number of special seminars. The book was written by one of the leaders of such a seminar (in Princeton) R. Pale in collaboration with other scientists. It contains a complete description of the solution to the index problem, including all the necessary topological and analytical apparatus. The book is intended primarily for topologists, but mathematicians of other specialties will find in it a lot of interesting things. It is also useful for teachers, graduate students and senior students of universities and pedestals.