Motivic integration is a powerful tool in algebraic geometry for studying singularities. The theory was conceived by Kontsevich in 1989 to provide a shorter proof of Batyrev's theorem. In 2009, a more "modern" form of this theory has emerged, spearheaded by Cluckers and Loeser. First, I'll talk about p-adic integration and motivations. Then I'll try to introduce the theory's basic objects, such as model theory and Grothendieck groups. Finally, if time permits, I'll set out some axioms of the motivic integral.