Mewomo, Oluwatosin Temitope.Olanipekun, Peter Olamide.2018-10-022018-10-0220172017http://hdl.handle.net/10413/15533Master of Science in Mathematics, Statistics and Computer Science. University of KwaZulu-Natal, Durban 2017.It has been realized that the definition of amenability given by B. E. Johnson in his Classical Memoir of American Mathematical Society in 1972 is too restrictive and does not allow for the development of a rich general theory. For this reason, by relaxing some of the constraints in the definition of amenability via restricting the class of bimodules in question or by relaxing the structure of the derivations, various notions of amenability have been introduced after the pioneering work of Johnson on amenability in Banach algebras. This dissertation is focused on six of these notions of amenability in Banach algebras, namely: contractibility, amenability, weak amenability, generalized amenability, character amenability and character contractibility. The first five of these notions are studied on arbitrary Banach algebras and the last two are studied on some classes of Segal algebras. In particular, results on hereditary properties and several characterizations of these notions are reviewed and discussed. Indeed, we discussed the equivalent of these notions with the existence of a bounded approximate diagonal, virtual diagonal, splitting of exact sequences of Banach bimodules and the existence of a certain Hahn-Banach extension property. Also, some relations that exist between these notions of amenability are also established. We show that approximate contractibility and approximate amenability are equivalent. Some conditions under which the amenability of the underlying group of a Segal algebra implies the character amenability of the Segal algebras are also given. Finally, some new results are obtained which serves as our contribution to knowledge.en-ZATheses - Pure Mathematics.Segal algebras.Banach algebras.Hahn-Banach.Amenability Banach algebras.Thesis