Noncommutative Geometry in the Framework of Differential Graded Categories

Also published as a journal article: Progress in Mathematics, 2010, 279: 253-275In this survey we discuss a framework of noncommutative geometry with differential graded categories as models for spaces. We outline a construction of the category of noncommutative spaces and also include a discussion on noncommutative motives. We propose a motivic measure with values in a motivic ring. This enables us to introduce certain zeta functions of noncommutative spaces.Snigdhayan Mahant