Abstract: In the first of a series of talks, I would like to introduce the notions of a Clifford algebra of a vector space $V$ over $\mathbb{R}$ and of a spin structure on a Riemannian manifold. I will discuss when a Riemannian manifold does in fact carry a spin structure, thus allowing it to admit spinors. This is not always possible because there may be topological obstructions on the manifold that inhibit it from carrying such a structure. Nevertheless, spin manifolds are useful for determining whether or not an orientable Riemannian manifold admits spinors. Once this is in place, we will look at the Dirac operator associated to a spin module and some of its properties, including how it operates on sections of the spinor bundle.