Unitarity is a fundamental property of quantum mechanics which underlies the dynamics of closed quantum many-body systems, the concept of thermalisation and the emergence of statistical mechanics. A different paradigm for quantum dynamics arises in presence of an external environment, which can represent for example dissipation due to a bath or an external monitoring apparatus[1,2]. Other sources of non-unitarity can arise for example in presence of non-Hermiticity due to post-selection of measurement outcomes [3]. The goal of this project is to explore the consequences of non-unitarity on the dynamics of quantum many-body systems, in particular for what concerns the dynamics of quantum information. Examples include: the study of entanglement dynamics in presence of quantum measurements or continuous monitoring, the use of measurements and active feedback to steer and prepare quantum many-body states.