Abstract. We survey some aspects of the theory of noncommutative manifolds focusing on the noncommutative analogs of two-dimensional tori and low-dimensional spheres. We are particularly interested in those aspects of the theory that link the differential geometry and the algebraic geometry of these spaces. These notes are based on a series of lectures given at the International Workshop on Noncommutative Geometry held at Tehran in September 2005. The aim of the lectures was to introduce some examples of noncommutative manifolds as well as some of the tools and notions needed for their study. We focused on those examples