Simulating the low-Re hydrodynamics of particulate flows is an extremely challenging and important problem that arises in several disciplines. In this talk, I will present recent advances made by our group in overcoming several computational bottlenecks, especially those arising in the context of dense suspensions confined by complex geometries. In particular, a spectrally-accurate scheme to resolve the interactions of close-to-touching particles, a novel periodizing scheme for arbitrary geometries and a new boundary integral equation formulation for colloidal and active suspensions will be presented. Incorporating stable time-marching schemes, fast direct solvers based on low-rank factorizations and the fast multipole method, we were able to simulate the hydrodynamics of over 1000 deformable particles flowing through a periodic microfluidic-chip geometry in less than a minute per time-step on a laptop. I will discuss several applications and our ongoing efforts to simulate the electrohydrodynamics of vesicle suspensions, evaluate the stresses experienced by motile cells, investigate the controllability of low-Re swimmers and to design microfluidic chips for cell sorting and separation.