Hydrodynamics is one of the oldest research areas in physics, with applications across all macroscopic scales in the Universe. Despite the long history of successes, however, fluid modelling still presents severe conceptual and computational challenges. Not surprisingly, the hurdles become even more formidable for relativistic flows, and new issues come to the fore too. This work is concerned with advancing multi-fluid models in General Relativity, and in particular focuses on the modelling of dissipative fluids and turbulent flows. Such models are required for an accurate description of neutron star phenomenology, and binary neutron star mergers in particular. In fact, the advent of multi-messenger astronomy—started with the first detection of a binary neutron star coalescence in 2017—offers exciting prospects for exploring the extreme physics at play during such cosmic fireworks. In this work we first focus on modelling dissipative fluids in relativity, and explore the arguably unique model that is ideally suited for describing dissipative multi-fluids in General Relativity. Modelling single fluids in relativity is already a hard task, but for neutron stars it is easy to argue that we need to understand even more complicated settings: the presence of superfluid/superconducting mixtures, for example, means that we need to go beyond single-fluid descriptions. We then consider turbulent flows and focus on how to perform “filtering” in a curved spacetime setting. We do so as most recent turbulent models in a Newtonian setting are based on the notion of spatial filtering. As the same strategy is beginning to be applied in numerical relativity, we focus on the foundational underpinnings and propose a novel scheme for carrying out filtering, ensuring consistency with the tenets of General Relativity. Finally, we discuss two applications of relevance for binary neutron star mergers. We focus on the modelling of (β-)reactions in neutron star simulations, and provide a discussion of the magnetorotational instability that is suited to highly dynamical environments like mergers. We focus on these two problems as reactions are expected to source the dominant dissipative contribution to the overall dynamics, while the magneto-rotational instability is considered crucial for sustaining the development of turbulence in mergers.