One approach to investigate the structure of an algebraic variety X is to study the geometry of curves, especially the rational curves, that X contains. This approach relies on classical geometric ideas and strives to understand the intrinsic geometry of varieties. It is nowadays understood that if X contains many rational curves, then their geometry determines X to a large degree.

After Shigefumi Mori showed in his landmark works that many interesting varieties contain rational curves, their systematic study became a standard tool in algebraic geometry. The spectrum of application is diverse and covers long-standing problems such as deformation rigidity, stability of the tangent bundle, classification problems, and generalizations of the Shafarevich hyperbolicity conjecture.

The talk concentrates on examples and basic properties of minimal degree rational curves on projective varieties. Some of the more advanced applications will be discussed.