This book addresses the rapid expansion of quantum mechanics, both in its theoretical development and in its wide-ranging applications across modern physics.The fields of quantum optics, quantum computers, quantum chemistry, quantum condensed matter, quantum electronics provide a flow of new phenomena, new theoretical concepts, new chapters in the formalism and modern often umexpected extentions of the old approaches. That brings the demand of what to teach in advanced courses of quantum mechanics. These are courses which follow the usual first two semesters of teaching the concepts and the formalism of the basic quantum mechanics.The book is based on extended lecture notes for such a course taught as a mandatory QM1 course in the physics graduate program. Together with the second QM2 course and advanced Statistical Physics the goal was to prepare the students for special courses in the 2nd semester which provided introduction to various research fields. The special distinct feature of this QM course was that its content was intended for all physics graduates including those who did not study relativistic extentions of QM.The above reasons were apparently behind growing popularity of the course with the students which let to an impressive increase of the number of students attending it in the recent years.