Free-surface flows -- flows that involve interfaces between multiple fluid phases -- are ubiquitous in many important technology applications such as coating, ink-jet printing and rocket propulsion as well as natural phenomena such as rain formation and aerosol emission during human expiratory event. These processes frequently involve the use of complex fluids — shear-thinning or viscoelastic non-Newtonian liquids, which often contain surfactants either as additives or as contaminants. Numerical simulations of these flows are challenging because the location of the free surface is unknown prior to the knowledge of the flow field, which cannot be prescribed at the outset and must be computed instead as part of the solution. In this talk, several examples of typical free surface flows will be discussed including coalescence of drops and bubbles, pinch of liquid filaments and bubbles, and perforation of a fluid sheets. The focus will be given to how bulk rheology and the presence of insoluble surfactant affect the dynamics of flows.