The famous Kontsevich´s formality theorem implies that equivalence classes of star-products on a real smooth manifold are in bijection with equivalence classes of formal Poisson structures. An equivalence class of a formal Poisson structure corresponding to a given star-product is often called the characteristic class (or Kontsevich´s class) of this star-product. In my talk I will discuss Morita equivalent deformations of the algebra of complex-valued smooth functions on a real differentiable manifold. I will explain how one can describe Morita equivalent star-products in terms of their characteristic classes. My talk is based on a joint paper with Henrique Bursztyn and Stefan Waldmann. It should be accessible to graduate students familiar with rudiments of differential geometry and with basic abstract algebra.