Several science and engineering applications involve fluid flows that cannot be accurately modelled through the standard, Newtonian Navier-Stokes equations. Some examples are blood, viscoelastic materials and multi-phase systems. Whenever fluid properties such as viscosity or density can vary in space and time, numerical challenges arise from both the algorithmic and theoretical sides. That includes, for instance, additional non-linearities, numerical instabilities, undesired matrix fill or loss of accuracy. Many of those issues can be overcome through the careful design of tailor-made numerical techniques such as stabilisation methods, semi-implicit (IMEX) stepping, shock capturing, advanced preconditioning, among others. This Mini-Symposium aims to bring together researchers in computational mechanics, scientific computing and numerical mathematics working on novel numerical methods for complex flows. Some examples of relevant problems and models are (1) fluids governed by complex rheologies (e.g., generalised Newtonian, viscoelastic or viscoplastic), (2) multi-phase or phase-change flows, (3) dispersed multi-fluid systems, (4) turbulent viscosity models, (5) thermally coupled flows, (6) diffuse-interface models (e.g. Cahn-Hilliard or Allen-Cahn), among others. Contributions are welcome within any class of numerical methods, addressing from the spatial to the temporal discretisation, from a priori analyses to solvers.