Popular growth models include:
* KPZ equation
* Dimer model
* Eden growth model
* SOS model
*Self-avoiding walk
*Abelian sandpile model
*Kuramoto–Sivashinsky equation (or the flame equation, for studying the surface of a flame front)
They are studied for their fractal properties, scaling behavior, critical exponents, universality classes, and relations to chaos theory, dynamical system, non-equilibrium / disordered / complex systems.
Popular tools include statistical mechanics, renormalization group, rough path theory, etc.