The theory of $p$-adic numbers is still relatively recent, but its applications have become numerous and widespread in the last half-century. In addition to their ubiquity in number theory, $p$-adic numbers and $p$-adic analysis provide a beautiful blend of algebra and analysis. The purpose of this talk is to discuss the construction of the $p$-adic numbers, their algebraic closure, and their subsequent completion. We'll then see how the metric properties of the field make certain questions much easier to solve, while severely limiting others. Finally, we'll discuss two simple applications of $p$-adic analysis in the forms of the Newton Polygon and Strassman's Theorem. If time permits, we'll also go over the properties of finite field extensions of $Q_p$.
Prereqs: Elementary analysis and algebra would be helpful. Also, a familiarity with some Galois theory would be great, but otherwise you'll just have to trust in some "hand-waving" arguments.