Accurate, efficient and robust solvers for complex-rheology flows remain a frontier topic in computational mechanics. This minisymposium concentrates on methodological advances and comparative studies, providing a venue for developers and analysts to exchange cutting-edge ideas, benchmarks and implementational details. Topics of interest include, but are not limited to • Variational, finite-element, finite-volume, spectral and lattice-Boltzmann formulations for viscoelastic, thixotropic, yield-stress and shear-thinning fluids • Stabilisation strategies for the high-Weissenberg-number and creeping-flow limits (log-conformation, DEVSS-G/SUPG and DG schemes), solvers for integral constitutive equations • Coupled micro–macro and multiscale frameworks • Data-driven and machine-learning approaches • Physics-informed neural networks (PINNs) and operator-learning surrogates • Hybrid solvers that embed ML modules within structure-preserving numerical cores • Reduced-order and latent-space models enabling digital-twin applications • Data-driven discovery of material laws using ML approaches • Sensitivity analysis, adjoint-based optimisation and uncertainty quantification in non-Newtonian settings • Stability, bifurcation and nonlinear wave phenomena in complex-fluid flows • Industrial and biomedical case studies: injection/compression molding, polymer extrusion, film coating, additive manufacturing, inkjet printing, blood rheology, soft-robot actuation, micro-fluidic devices By bringing together experts on algorithmic theory, high-performance implementation and application-driven validation, this minisymposium aims to accelerate next-generation numerical technologies for complex fluids. We invite contributions from academia and industry, and particularly welcome studies that promote reproducibility and synergy across different methodological communities.