This talk will serve as a more mathematically-oriented companion to the talk It's Hard to Choose: An Introduction to the Mathematics of Voting. We will begin with a simple and elegant proof of Arrow's Impossibility Theorem. Next, we will look into some of the paradoxes involved in positional voting procedures. (This part is based on joint work with Jackie Anderson and Brian Camara.) We will conclude with a brief look at voting rules based on minimizing distances in the symmetric group.