This paper deals with two aspects of the theory of characteristic classes of star products: first, on an arbitrary Poisson manifold, we describe Morita equivalent star products in terms of their Kontsevich classes; second, on symplectic manifolds, we
describe the relationship between Kontsevichs and Fedosovs characteristic classes of star products.
After an overview of noncommutative differential calculus, we construct parts of it explicitly and explain why this construction agrees with a fuller version obtained from the theory of operads.
