Post-quantum cryptography is a rapidly growing area of cryptography which encompasses cryptographic algorithms that we can implement now, on current devices, but which are designed to be secure against future scalable quantum computers. In this talk, we will present some of the computational number theory problems that are at the core of post-quantum cryptography. This includes the ideal shortest vector problem, on finding small elements in ideals of orders in very high degree number fields, as well as the computational supersingular isogeny problem, on computing an isogeny between two given supersingular elliptic curves. We will also discuss some recent advances on these problems, including computational work interrogating the randomness of supersingular isogeny graphs. This includes joint work with Ross Bowden and Felix Marsh.