The flows of complex fluids link fundamental research questions to potential applications, both in industry and for understanding natural phenomena. In this talk I discuss three research questions that we have studied recently: (1) Although flows at modest Reynolds numbers at a T-shaped junction is a geometry where one should expect everything is known, nevertheless we uncover previously unrecognized complexity in three-dimensional solutions to the Navier-Stokes equations, which rationalize our experimental observations of particle trapping in this common flow configuration. (2) A spherical particle translating at low Reynolds numbers (slow flows) parallel to a rigid wall maintains the same separation distance during its motion. However, we show using theory and experiments that a particle moving along an elastic membrane, which can deform by bending, is repelled from the membrane due to hydroelastic forces. (3) Thermal gradients in fluid systems can lead to the formation of distinct layered patterns. We report experiments injecting a fluid into a second miscible phase and show that, above a critical injection velocity, layering emerges over a time scale of minutes; the same phenomenon can occur in café latte.