The incompressible Navier-Stokes equations are infused with important physical structure, evidenced by a wide array of balance laws for momentum, energy, vorticity, enstrophy, and helicity. The key to unlocking this structure is precisely the volume-preserving nature of incompressible flow, yet most numerical methods only satisfy the incompressibility constraint in an approximate sense. Consequently, such methods do not obey certain fundamental laws of physics and produce unphysical results for many flow configurations of interest. In this talk, I will discuss recently developed spline discretizations which satisfy the incompressibility constraint point-wise and hence replicate the balance law structure of the Navier-Stokes equations. I will give a brief overview on how to construct such discretizations, discuss their relationship to standard finite element, finite volume, and spectral methods, and present numerical results illustrating their promise in the context of low-speed flows.