"This proposal concerns open systems, i.e. systems interacting with the environment, and their fundamental role in natural sciences. The main objectives are: i) to develop theory of Brownian motion for molecules in biological environments; ii) to adapt classical many-body open systems such as kinetic or/and diffusion-aggregation models to the quantum domain; iii) to develop theory of open systems as quantum simulators; finally iv) to develop theory of quantum Brownian motion in inhomogeneous media. Although all these objectives may seem to be quite unrelated, our main goal will be to connect them in order to unambiguously asses the relevance of open systems in specific areas of physics, biology and beyond. Accordingly, objective i) will be explored in close collaboration with experimentalists in which the diffusion of biomolecules on cell membranes requires a description in terms of Brownian motion in correlated disordered potentials. In ii) we will search for many-body kinetic and growth models that provide the configurations that may serve as samples of random potentials desired in i). These models can be regarded as quantum models with non-Hermitian generators of evolution; in some situations they can be generalized to genuine quantum ones, described by a quantum master equation, linking ii) and iii). In iii) we will look for applications of quantum open systems as quantum simulators of condensed matter/high energy physics. We will also look at single particle interactions with quantum many body environment, linking the objectives iii) with iv) and i). Expected results are: a) understanding the relationship between biological function and the spatiotemporal dynamics of single molecules in living cells; b) understanding of the structure of classical many body stochastic models and their relation to quantum ones; c) concrete proposals for open systems quantum simulators; and d) development of tools to characterize and observe quantum Brownian motion."