In this talk I will develop the main aspects of quantum cohomology for homogeneous spaces. After a quick reminder of the main properties of quantum cohomology, I will explain why homogeneous spaces provide a particularly nice family of examples for Gromov-Witten theory. In particular, I will explain why their quantum cohomology is enumerative. This has interesting consequences on some traditional problems in enumerative geometry, such as for the counting of plane curves subject to certain incidence conditions (Konstevich’s formula for plane curves). Finally, restricting myself to Grassmannians for simplicity (and to small quantum cohomology), I will introduce the basic properties of “quantum Schubert calculus”, including the “quantum-to-classical principle” of A.S. Buch relating Gromov-Witten in homogeneous spaces to classical intersection numbers in an auxiliary space.