In 1998, Hastings and Levitov proposed a family of models for random growth, indexed by a parameter alpha, which includes versions of diffusion-limited aggregation (DLA) for mineral deposition and the Eden model for biological cell growth. They predicted a change at alpha = 1 (which corresponds to the Eden model), from stable to turbulent behaviour. In this talk, I will show that the limit dynamics of the HL model, as the particle size goes to zero, follow solutions of a certain Loewner-Kufarev equation, where the driving measure is made to depend on the solution and on the parameter alpha. The fluctuations around the scaling limit are shown to be Gaussian, with independent Ornstein-Uhlenbeck processes driving each Fourier mode, which are seen to be stable if and only if alpha is less than or equal to 1, consistent with Hastings and Levitov's prediction. This talk is based on the paper https://arxiv.org/abs/2105.09185 which is joint work with James Norris and Vittoria Silvestri.