The Macdonald polynomials are a homogeneous basis for the algebra of symmetric polynomials, which generalize many important families of special functions, such as Schur polynomials, Hall-Littlewood polynomials, and Jack polynomials.

The interpolation polynomials, introduced by F. Knop and the speaker, are an inhomogeneous extension of Macdonald polynomials, which are characterized by very simple vanishing properties.

The binomial coefficients are special values of interpolation polynomials, which play a central role in the higher rank $q$-binomial theorem of A. Okounkov.

We will give an elementary self-contained introduction to all three objects, and discuss some recent results, open problems, and applications.