In the first part of this talk my goal is to convey the importance of mass conservation in fluid flow simulations in an accessible manner. Mass conservation will be seen as one of two important structure preservation properties that are desirable for Stokes flow approximations. It is related to the proper treatment of the incompressibility constraint on the fluid velocity u, namely div(u)=0. One can formulate this condition in different Sobolev spaces, resulting in different categories of methods. In the second part of the talk, focusing on methods that yield fluid velocities in the Sobolev space H(div), we seek a natural Sobolev space for viscous fluid stresses. We report on our research findings on a mixed formulation that produces, in addition to exactly divergence-free velocity approximations, good approximations to viscous stresses. The main new insight is that stresses should lie in a nonstandard Sobolev space H(curl div). For the finite element enthusiast, we will outline, time permitting, the construction of a new finite element family, useful for implementing matrix fields with continuous normal-tangential components. This new finite element space is used to approximate fluid stresses.