The paper is, as its title states, a “minicourse” on some aspects of Poisson geometry. The author focuses on results obtained by V. Dolgushev [Sel. Math., New Ser. 14, No. 2, 199–228 (2009; Zbl 1172.53054)], but other subjects are also touched. After providing some preliminaries on Poisson geometry, the author introduces the notion of modular class of Poisson manifold, and exemplifies its importance. Then he switches to deformation quantization framework and summarizes Dolgushev’s results. They can be roughly described in the following way. Each deformation quantization algebra comes with a distinguished automorphism (up to inner one), which appears in van den Bergh dualizing bimodule. Using Kontsevich formality this automorphism can be written quite explicitly, and for the case of unimodular Poisson manifold it turns out to be inner. The author comments on the relation between deformation quantization and noncommutative geometry à la Connes, and stresses importance of unimodularity in building noncommutative trace (compare also G. Felder and B. Shoikhet [Lett. Math. Phys. 53, No. 1, 75–86 (2000; Zbl 0983.53065)]). Finally, the special case of Poisson-Lie groups is described in some detail.