In this paper, we present Riemann zeta function and its analytic continuation and functional equation. We will begin with the Gamma function and some properties of the Gamma function. Then we prove that the Riemann zeta function can be represented by Euler’s product. We will present proof of analytic continuation of the zeta function using Gamma function. Also, we define a Jacobi Theta function and Xi function with the proof of its functional equation and then it will be used to prove the analytic continuation of zeta function as well as its functional equation.