This paper deals with alternative cryptosystems which resist the attacks of quantum computers. The work is divided into 3 chapters. In the first chapter we give the motivation to observe such systems. We explain some basic concepts from cryptography and point out some key features that make a given cryptosystem good. Then we give a short introduction to quantum computing, explaining the difference between a classical and a quantum computer, in addition to showing how computation works in a quantum environment. For the end of the first chapter we briefly explain the most important quantum algorithm: Shor’s algorithm. Shor’s algorithm allows us to easily factorize numbers and thus break most of the cryptosystems used today. For this reason, there is a need for cryptosystems based on a different class of problems. In the second chapter we present the first class of such problems, the problems defined over lattices, and we explain an example of a cryptosystem based on these problems, the GGH cryptosystem with a public key. At the beginning this chapter, we define lattices and show some key features. Then we describe most of the major problems on lattices such as SVP and CVP. To conclude, we define the GGH cryptosystem, explain how it works and what are the drawbacks and advantages. In the last, third chapter, we present another class of problems that are currently resisting quantum attacks, code based problems. First we explain what the codes are and using an example code explain their essential properties. The most important of all these features is the ability to correct errors and precisely around this feature has Mceliece designed his cryptosystem. At the end of the chapter, we explain the idea of the Mceliece cryptosystem and comment on its quality for practical application.