The placenta is a multifunctional organ that exchanges blood gases and nutrients between a mother and her developing fetus. In humans, fetal blood flows through intricate networks of vessels confined within villous trees, the branches of which are bathed in pools of maternal blood. Fluid mechanics and transport processes play a central role in understanding how these elaborate structures contribute to the function of the placenta and how their disorganization may lead to disease. Recent advances in imaging and computation have spurred significant advances in simulations of fetal and maternal flows within the placenta across a range of length scales. Models describe jets of maternal blood emerging from spiral arteries into a disordered and deformable porous medium, as well as solute uptake by fetal blood flowing through elaborate three-dimensional capillary networks. We survey recent developments and emerging challenges in modeling flow and transport in this complex organ.