Fluid flows in or around modern nanodevices can attain extreme parameters. In this parameter space, the Navier-Stokes equations, which describe everyday fluid dynamics and are perhaps the most important set of equations in all of engineering, can sometimes fail. The failure of the Navier-Stokes equations in nanodevices can take place via different physical mechanisms: for instance, the continuum hypothesis may break down due to a size effect; or the local equilibrium may be violated due to the high rate of strain. In this talk, I will first introduce the Navier-Stokes equations and describe a framework for understanding how they break down in nanoflows. I will then present our intuitive experiments based on nanomechanical resonators (shown in SEM image), which provide a rigorous test for the fundamental ideas. In the approach to nanofluidics, a single dimensionless scaling parameter, which combines length and time scales in the flow, emerges as the dominant scaling parameter. This parameter may be of relevance for understanding and predicting a number of other nanoflows of technological importance.