There is a lot of interest in characterizing the mechanics of complex interfaces that compose biological systems such as cells. In this talk, we discuss some of our recent work on the micromechanics of vesicles, i.e., sacs of fluid of ~20 microns containing a phospholipid bilayer. Here, we focus on how these systems behave in general linear flows, probing the conditions under which they become mechanically unstable and form thin threads. We find that vesicles exhibit qualitatively different shape transitions than droplets under flow due to the bending and dilatational resistance of the phospholipid bilayer. We discuss the microfluidic experiments and boundary element simulations to quantify the different shape transitions, and describe how flow type and flow history alters these dynamics. In the second half of the talk, we discuss more general problems on how shear and dilatational resistance of a membrane alter the dynamics of droplet-like systems found in biology. We develop analytical theories to quantify how linear shear and dilatational surface moduli alter droplet translation, shape, breakup, and particle lift. We find that one can use simple symmetry arguments to illuminate how interfacial shear viscosity alters the translational speed of a droplet, and one can use simple physical arguments to understand how interfacial effects alter droplet breakup.