In this thesis we consider four families of nonlinear recurrences which can be shown to either fit into Fomin and Zelevinsky's framework of cluster algebras, or the more general setting of Laurent phenomenon algebras given recently by Lam and Pylyavskyy. It then follows that each family of recurrences we study possesses the Laurent property. Our main interest lies in the linearisability and Liouville integrability of the maps defined by these families. We prove that three of the families are linearisable. Firstly, we study examples arising in the context of cluster algebras and provide a detailed survey of recent results of Fordy and Hone, with the aim to develop the understanding of Liouville integrability for odd order examples of this type. Following this, we extend the results of Heideman and Hogan, to show that their family of nonlinear recurrences is linearisable for general initial data. The third order example from this family of recurrences admits a different generalisation of a new family of nonlinear recurrences for which we also show the general case to be linearisable. We also present a connection with the dressing chain which provides a generating function for the first integrals for recurrences of this type. Lastly we study a family of Somos-type recurrences which is not linearisable. However we present the method of finding the Lax representation from which we can generate first integrals and show that the examples of recurrences studied here, arising in the context of cluster algebras, are Liouville integrable.