The concept of a group helps to unify a great variety of different mathematical structures which at first sight might appear unrelated. In this chapter we start by looking at groups of permutations from which the concept of a group took its origin. Permutations have a diverse range of applications to cryptography. We pay a special attention to orders of permutations and analysis of repeated actions. We briefly consider several topics in general groups such as isomorphism, subgroups, cyclic subgroups, orders of elements. Lastly, we consider the group of points of an elliptic curve and explain the basics of the elliptic key cryptography and Elgamal cryptosystem.