Abstract:  Public key cryptography is an indispensable part of our modern communication systems. However, quantum computers can defeat cryptosystems like RSA, which are based on ``hard" number theory problems. Recently a great effort has been put into the search for alternative public key cryptosystems. The family of multivariable public key cryptosystems (MKPC) is one such promising alternative. Their theoretical security assumption is the proven theorem that solving a system of polynomial equations over a finite field is in general NP-hard and quantum computers are not yet shown to be effective in solving this problem. Furthermore, computations in a finite field can be more efficient than manipulating larger and larger numbers, as required by the systems based on number theoretical problems. In this talk, we will present the development in this area, mainly we will talk about the Matsumoto-Imai, the Sflash, the HFE (Hidden Field Equation), the HFEv, the Oil-Vinegar, the TTM and the internal perturbation construction, and the attacks on those cryptosystems.