Many important phenomena reveal stochastic geometrical objects and shapes. Among them are fluctuating domain boundaries in statistical mechanics, growing patterns in non-equilibrium processes, and fluctuating surfaces studied in random matrix theory. These geometrical objects naturally arise in the theory of 2D growth processes, disordered systems and random media. In many interesting cases they are fractal in nature. The project focuses on a wide class of processes involving stochastic geometry in two dimensions and the related deterministic objects arising in free-boundary problems, such as Laplacian and elliptic growth. In spite of discovery of many deep connections between the theory of moving interfaces in two dimensions to a number of modern branches of mathematics such as advanced complex analysis, deformations of Riemann surfaces, integrable systems and theory of random matrices, there are many important questions to be addressed. For instance, complete analytic description, classification and universality of random growth processes and their deterministic counterparts on the plane as well as theory of singularity formation and regularisation are far from being complete. The project goal is to apply novel analytical and numeric techniques and combine ideas from different disciplines, in order to attack the above problems. Remarkable developments in Laplacian and elliptic growth due to recent achievements in theory of integrable systems and random matrices as well as revitalization of the study of 2D critical phenomena as a stochastic evolution of geometry due to recent discovery of the Stochastic Loewner Evolution make feasible further significant advances in the field. Multi-disciplinarity of the present project is addressed to combine the most recent advances in the named adjacent topics to shed light on the nature of fascinating interaction amongst phenomena both of pure physical and mathematical origin.