This thesis is concerned with finitely-generated ideals in certain Banach algebras, where “finitely-generated” is understood in the following sense: Let A be an algebra and A# its unitisation. A left ideal I in A is finitely-generated if there exist n ? N and x1,...,xn ? A such that I = A#x1 #43;··· #43; A#xn. In the case that a left ideal infinitely-generated, it is immediate that the generators all belong to the ideal. Note also that when this definition is applied to topological algebras we do not take the closure on the right-hand side. The Banach algebras of most interest to us will be weighted group algebras and weighted semigroup algebras, so next we define these terms precisely.