The building blocks of mathematical morphogenesis were put several decades ago in the seminal works of Turing and Eden. Their goal was understanding how a macroscopic structure, in particular one breaking the initial homogeneity, could arise out of a multiplicity of simple interactions. While the approach of Turing implied the use of reaction-diffusion equations, Eden concentrated on a probabilistic abstraction of a developing cell colony. In particular, he studied the architecture of a lattice cell colony to which new cells were added following certain probabilistic rules. The objective was studying the asymptotic colony profile. The original Eden problem can be greatly generalized by means of the use of stochastic partial differential equations. They allow a systematic study of the properties of the colony periphery, particularly of the interface fluctuations. In this talk we will summarize our recent progress in this field, concentrating on the properties of the realizations of the stochastic growth process. Our goal is unveiling under which conditions the developing radial cluster asymptotically weakly converges to the concentrically propagating spherically symmetric profile.