Holomorphic functions are the main concern of complex analysis. The Cauchy-Riemann Equations is one of the many tools to study holomorphic functions. We will take a look at these equations in one and in two complex variables. We will present an application of the Cauchy-Riemann Equations in $\C^1$ to provide an idea about the role it plays in higher complex dimensions. The seminar will be accessible to graduate students.