Multivariate Public Key Cryptosystem (MPKC) seem to be promising toward future digital security even in the presence of quantum adversaries. MPKCs derive their security from the difficulty of solving a random system of multivariate polynomial equations over a finite field, which is known to be an NP-hard problem. This article aims at presenting a comprehensive survey that covers multivariate public key encryption and signature schemes specifically targeting toward security, efficiency, and parameter choice. The survey starts by giving an overview of the existing security challenges which include structural attacks such as MinRank attack, differential attack , and finding Gröbner basis for direct attack , and so on. Additionally, it discusses the necessary algorithms for the implementation of the multivariate schemes. This study also compares the promising multivariate encryption and signature schemes. The critical open challenges that are reviewed in this survey will serve as a single comprehensive source of information on multivariate encryption and signature schemes and a ready reference for researchers working in this rising area of public key cryptography.