Recent advances in Brill--Noether theory and the geometry of Brill--Noether curves

The first goal of this article is to survey recent progress in Brill--Noether theory, including both the study of the moduli space of maps from a curve to projective space and the geometry of the resulting curves in projective space. The second goal is to introduce newcomers to some of the important techniques that have been introduced or developed in the last decade that made these advances possible.