Earlier this semester, Michael Shapiro spoke in the combinatorics seminar about double Hurwitz numbers: consider this talk a retroactive introduction and motivation to this field. Hurwitz numbers are combinatorial interesting objects in their own right and are related to the symmetric group, but are also important tools in studying the moduli space of curves. We hope to give a flavor of both the symmetric group and algebraic geometric approaches, and their interplay. If time permits, we will wave our hands at Goulden, Jackson and Vakil's proof that Double Hurwitz numbers are piecewise polynomial.