The main topics of interest in this thesis will be two types of complexity, abelian complexity and permutation complexity. Abelian complexity has been investigated over the past decades. Permutation complexity is a relatively new type of word complexity which investigates lexicographical ordering of shifts of an aperiodic word. We will investigate two topics in the area of abelian complexity. Firstly we will consider an abelian variation of maximal pattern complexity. Secondly we consider an upper bound for words with the C-balance property. In the area of permutation complexity, we compute the permutation complexity function for a number of words. A formula for the complexity of Thue-Morse word is established by studying patterns in subpermutations and the action of the Thue-Morse morphism on the subpermutations. We then give a method to calculate the complexity of the image of certain words under the doubling map. The permutation complexity function of the image of the Thue-Morse word under the doubling map and the image of a Sturmian word under the doubling map are established.
