In this colloquial talk I shall present a few problems arising in various applied fields, that give rise to partial differential equations in a priori unknown domains; thus the term free boundary. These problems in general include phase transitions such:
i) the solid-liquid interphase when the temperature goes through the solidification of some material,
ii) the boundary between exercise and continuation region for a financial instrument (such as option, and optimal stopping),
iii) the transition from elastic to plastic behavior when stress goes through a critical value,
iv) the diffusion limited aggregation (controlled random walks on lattice and their continuous limits).
I shall keep the talk at non-technical level, and will only explain the problems from various view points, stressing on the mathematical models and their formulations. A few results, tools and methodology will be mentioned.