Multivariate public key cryptography is one of the main families of post-quantum cryptography, offering fast and compact schemes built from systems of polynomial equations over finite fields. In this mini-course, we’ll begin with an introduction to past and present multivariate cryptosystems. From there, we’ll explore the MinRank problem, a central tool in cryptanalysis that has been used to break several high-profile schemes in both the multivariate and code-based regimes. This course aims to make these ideas accessible to participants with a basic background in algebra, while offering a window into current research directions in post-quantum cryptography.