Beta expansions are well known generalisations of the familiar integer base representations of real numbers. Importantly a real number x often has many beta expansions. As such, it is natural to ask whether a real number x has a beta expansion that satisfies a certain additional property. Properties we are interested in may relate to digit frequencies, complexity, etc. In this talk I will survey a number of results in this direction and provide a flavour of their proofs. I will also pose some open questions.