In this talk we will have an introduction to Kasparov's bivariant K-theory. We will start with a brief reminder about the notions of Fredholm operators, C^*-algebras and K-theory. Then, we will introduce elliptic operators over manifolds, which (likely) motivates the definition of KK-theory. Finally, we will talk about Kasparov product map and discuss several of its applications. If time permits, we will explore further topics related to the so-called assembly map and Baum-Connes' conjecture.