In this thesis we will be exclusively considering uncountable groups and semigroups. Roughly speaking the underlying problem is to find “large” subgroups (or subsemigroups) of the object in question, where we consider three different notions of “largeness”: we classify all the subsemigroups of the set of all mapping from a countable set back to itself which contains a specific uncountable subsemigroup; we investigate topological “largeness”, in particular subgroups which are finitely generated and dense; we investigate if it is possible to find an integer r such that any countable collection of elements belongs to some r-generated subsemigroup, and more precisely can these elements be obtained by multiplying the generators in a prescribed fashion.