Multivariate public key cryptosystems (MPKC) is a fast-developing area in cryptography. In the past 10 years, MPKC schemes have increasingly been viewed as a possible alternative to number theoretic-based cryptosystems such as RSA, as they are generally more efficient in terms of computational effort. As quantum computers are developed, MPKC will become a necessary alternative. "Multivariate Public Key Cryptosystems" systematically presents the subject matter for a broad audience. The first chapter introduces the basic ideas and early development of both multivariate public key cryptography and signature schemes. The next five chapters address the main families of multivariate schemes and the concept of perturbation, which is a means by which the security of various schemes can be improved without much cost in efficiency. Each family is introduced in terms of the origin of the material.