We want to discuss the development of K-theory and the motivation behind it. We will discuss the definitions of $K_0(R)$ and $K^0(X)$, where $R$ is a commutative ring with 1 and $X$ is a compact Hausdorff topological space. So it will give us both the algebraic and topological point of view of the development of K-theory and then we will try to see the equivalence between them.