Introduction The $\textbf{degree}$ of a function $f$ defined on a set $\Omega$ and its image $f\left(\Omega\right)$ is an extension of the winding number , which counts the number of times a closed curve travels counterclockwise around a given point. Intuitively, the degree counts the number of times $f$, in some sense, ''wraps'' $\Omega$ around a point in $f\left(\Omega\right)$. So what additional information does the degree give, and why would you care about it? The degree was developed as a way to measure, or keep careful count of, the number of solutions to a system of nonlinear equations. By ''careful'' we mean consistent with respect to some types of perturbation of the system. Degree theory, as we shall see shortly, provides some very nice tools for verifying the existence of solutions to nonlinear systems of equations. We will keep this conversation light, general and mostly geared towards imparting an intuition concerning wh...