  These are the lecture notes from my portion of a mini-course for the summer
school "Building Bridges 3" that was held in Sarajevo during July 2016. My
lectures covered the Katz definition of modular forms, a family of forms
defined from this perspective and their relation to Eisenstein series, and
methods of finding explicit models of modular curves. The treatment is purely
expository, and the results are mostly standard, although a few points of view
may not be as widely known as they deserve to be.
