Abstract: We will introduce the Clifford algebra associated to a vector space equipped a quadratic form. As an important case, then we give a description of the Clifford algebra for $\mathbb{R}^n$ and $\mathbb{C}^n$, equipped with the standard quadratic form, as a subalgebra of matrix algebras and prove the periodicity theorem for these algebras. Moreover, representation theory of the Clifford algebras will be discussed.