Circle action is easy to describe in topology. However, we also have serval ways to describe algebraic circle action that are not very intuitive. We will start with cohomology theories associated with a smooth circle action on a smooth manifold. In particular, we will focus on the equivariant de Rham cohomology. Then we will see where the notion of algebraic circle action comes from. Finally, we can construct the algebraic circle action of the Hochschild complex of associative algebra via this point of view.