This will be a nontechnical (beyond knowing something about curves) introduction to some of my favourite surfaces. Just as for curves, there is a classification statement and the different parts of the classification have different aspirations, techniques and results.

The Magma system has a lot of functions for working with algebraic curves. I will review the classification of curves and compare it to the Magma typing regime. Then I'll give an overview of the classification of surfaces to show how a similar typing system might work.