Conformal-map dynamics for a class of transport-limited growth phenomena Martin Z. Bazant, Jaehyuk Choi, Benny Davidovitch For diffusion-limited growth in two dimensions, remarkable theoretical progress has been made using time-dependent conformal maps. Continuous Laplacian growth, such as quasi-steady solidification or viscous fingering (where the pressure is harmonic), has been described in this way for over half a century, and recently Hastings and Levitov extended the approach to stochastic (discrete) Laplacian growth, such as diffusion-limited aggregation (DLA) and dielectric breakdown. Here, we generalize these powerful methods to a class of transport-limited non-Laplacian growth phenomena, focusing on growth from a finite seed by advection-diffusion in a uniform potential flow. For discrete growth, e.g. dilute-particle aggregation in a flow, we find a universal crossover from DLA to a new advection-dominated fractal-growth regime. For continuous growth, e.g. solidification in a flowing melt, an analogous dynamical crossover proceeds until cusp singularities form in finite time (without surface tension). In both cases, the scaling variable is a time-dependent effective Peclet number. .