Grötzsch's
theorem from 1959 says that every planar
triangle-free graph has chromatic number at most 3.
In this talk we discuss some problems and results that have been
inspired by Grötzsch's theorem. They include list colorings, extensions
to higher surfaces,
the 3-color space, the number of 3-colorings, and edge-decompositions
into a fixed tree. We also describe some of the proofs of Grötzsch's
theorem.