We'll talk about a central problem in algebraic geometry: classifying smooth projective varieties, motivating why we would care and why this is an incredibly difficult problem.  To do this, we'll introduce a fundamental tool in birational geometry--blow ups--and study them through many examples.  We'll define the notion of a birational map and see that blow ups are the only birational maps we need to care about.  Studying birational equivalence classes will lead into some classic moduli problems and a brief mention of singular varieties.  The talk will be full of many friendly polynomials and examples to bring intuition to this abstract topic!