Groups are mathematical structures in abstract algebra. They are a ordered pair (G, ·), in which · is a binary operation on G. In addition the group needs to closed under the operation, it needs to be associative and it needs to have both the neutral element and the inverse element for each element. The undergraduate thesis shows different types of groups and gives examples of them. The second chapter talks about groups in general. In it we have the definition of the group and the proposition about the uniqueness of the inverse element and the neutral element. We also have propositions for product inversion and the truncation rule. Proofs are given for all the propositions. Commutative groups (Abelian groups) are defined and a brief historical overview is given. Finally we have examples. The next chapter contains definitions and examples of subgroups. A way to determine the order of subgroups, if they exist, is given through the statement and proof of Lagrange’s theorem. The chapter ends with an example. In the fourth chapter definitions of cyclic groups and the elements that create them - generators are given. The chapter contains three examples of cyclic groups. The next chapter deals with cosets, normal subgroups, and quotient subgroups. The fifth chapter deals with means of comparing groups. Definitions and examples of homomorphisms and isomorphisms are given. The next chapter deals with the Cartesian product or the direct product of groups. A definition is given as well as an example of multiplication. In the penultimate chapter, the definition of the permutation group is given. Examples are given on the symmetric group S3. The last chapter deals with symmetry groups. Definitions of isometries and symmetry groups are given, supported by examples.