  In this tutorial, we provide an overview of many of the established
combinatorial and algebraic tools of Schubert calculus, the modern area of
enumerative geometry that encapsulates a wide variety of topics involving
intersections of linear spaces. It is intended as a guide for readers with a
combinatorial bent to understand and appreciate the geometric and topological
aspects of Schubert calculus, and conversely for geometric-minded readers to
gain familiarity with the relevant combinatorial tools in this area.
  We lead the reader through a tour of three variations on a theme:
Grassmannians, flag varieties, and orthogonal Grassmannians. The orthogonal
Grassmannian, unlike the ordinary Grassmannian and the flag variety, has not
yet been addressed very often in textbooks, so this presentation may be helpful
as an introduction to type B Schubert calculus.
  This work is adapted from the author's lecture notes for a graduate workshop
during the Equivariant Combinatorics Workshop at the Center for Mathematics
Research, Montreal, June 12-16, 2017.
