As we move into a new paradigm of computation, almost all of our assumptions about the security of our current cryptosystems are wrong. Large-scale quantum computers are known to be able to execute algorithms capable of efficient factoring and discrete logarithm computations. Unfortunately, most of the public-key cryptography widely used on the Internet today - including RSA and Elliptic Curve Cryptography - is based on the presumed hardness of these exact problems. In this talk, I introduce quantum computation and the practical realities it will have on popular cryptosystems - both technically, as well as from the perspective of a variety of use cases. From here, I introduce the new mathematics we're currently building to replace RSA in a post-quantum world, and the entirely new cryptanalytic tools we'll need to use to construct it.