Most classical and modern studies of swimming, pumping, and mixing in fluids have considered fluids that are Newtonian. All of these phenomena also take place in fluids that are viscoelastic and at low Reynolds number, and are particularly important to biology and to engineering areas such as microfluidics. I will discuss theoretical studies of the effect of viscoelasticity on low Reynolds number undulatory swimming and peristaltic pumping. I will also discuss an example of how symmetry breaking instabilities in extensional flows of a viscoelastic fluid can lead to new coherent structures and fluid mixing.