The Grothendieck ring of varieties over a scheme S is an interesting algebraic construction whose elements are finite type S schemes related by cut-and-paste. We will define this ring, explore some properties, define Motivic invariants with some examples, and explain at the end how these ideas were used in proving a theorem of Kontsevich concerning Hodge numbers of birational Calabi-Yau varieties, using an argument from p-adic integration due to Batyrev.