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 of the curious features of these models is
that although the models are constructed in an isotropic way, scaling limits appear
to be anisotropic. In this talk, we construct a family of models in which randomly
growing clusters can be represented as compositions of conformal mappings. We
are able to show rigorously that for certain parameter choices, the scaling limits
are anisotropic and we obtain shape theorems in this case. This contrasts with
earlier work on related growth models in which the scaling limits are shown to be
growing disks.