In 1879, Hermann Schubert published the celebrated "Kalkül der abzählenden Geometri" ("Calculus of Enumerative Geometry"), wherein he considered questions of the following form:

"Wieviel geometrische Gebilde von bestimmter Definition erfüllen gewisse gegebene Bedingungen?" (Schubert's emphasis), or roughly "How many geometric objects of certain definitions fulfill certain given conditions?"

A typical such question that will guide this talk is:

How many lines intersect four given general lines in (projective) 3-space?

The method used to solve this counting problem and others like it in enumerative geometry is known as Schubert calculus. Another counting problem amazingly solved by Schubert in this way was: "given twelve quadrics in $\mathbb{P}^3$, how many twisted cubics are tangent to all twelve?" Schubert (correctly) computed the answer to be $5 819 539 783 680$!!!

In this gentle introduction to Schubert calculus, we will begin with the theme: Schubert Calculus As Schubert Did It, before progressing to more modern variations of Schubert calculus, in particular via the cohomology of Grassmannians, that allow us to give Schubert's method a more rigorous foundation.