Planar random growth processes occur widely in the physical world. Examples include diffusion-limited aggregation (DLA) for mineral deposition and the Eden model for biological cell growth. One approach to mathematically modelling such processes is to represent the randomly growing clusters as compositions of conformal mappings. In 1998, Hastings and Levitov proposed a family of such models, which includes versions of the physical processes described above. In earlier work, Norris and I showed that the scaling limit of the simplest of the Hastings-Levitov models is a growing disk. Recently, Silvestri showed that the fluctuations can be described in terms of the solution to a stochastic fractional heat equation. In this talk, I will discuss on-going work with Norris and Silvestri in which we establish scaling limits and fluctuation results for a natural generalisation of the Hastings-Levitov family.