Stochastic generalizations of moving boundary problems appear quite naturally in the continuum description of e.g. solidification problems. Perhaps the simplest example is provided by a so-called one-sided solidification problem in which a condensed (solid) non-diffusing phase grows at the expense of a diluted diffusing phase (vapor or liquid). In this context, noise terms can be introduced to account for fluctuations in the interface kinetics leading to irreversible growth, and in the diffusive currents in the diluted phase. Thus, an effective closed evolution equation for the interface profile can be derived in a systematic way, carrying both deterministic and stochastic contributions, with parameters that can be related to those of the full moving boundary problem. This effective equation provides an interesting instance in which one can study the interplay between noise and non-local effects induced by diffusive interactions. Going beyond the approximations made in this process requires, e.g., formulation of a (stochastic) phase-field description that is equivalent to the original moving boundary problem. In turn, phase-field simulations allow to explore the rich morphological diagram that ensues. Applications will be discussed in the context both of non-living and biological systems subject to diffusion-limited growth, such as surfaces of thin films produced by Chemical Vapor Deposition or by Electrochemical deposition, or bacterial colonies. We will describe work done in collaboration with Mario Castro and Matteo Nicoli.