The concept of a groupGroup helps to unify a great variety of different mathematical structures which at first sight might appear unrelated. In this chapter we will start by looking at groups of permutations from which groups take their origin. We will then give a general definition of a group, and move on to studying the multiplicative group of \(\mathbb Z_n\) and the group of points of an elliptic curve. The latter two groups have recently gained cryptographic significance. Group theory plays a central role in cryptography; as a matter of fact, any large finite group can potentially be a basis of a cryptographic system.