I'll spend the first half (but hopefully less) of the talk giving an abridged introduction to the Euler and Navier-Stokes equations, covering the basic derivations and some of the known results. Afterwards, I will provide two examples to illustrate some of the difficulties in analyzing these equations. The first example is a 1-dimensional model problem for the 3-dimensional Euler equations that mimics turbulent behavior. This model is simpler, but fails global well-posedness in a bad way. The second example is an Eulerian-Lagrangian recasting of Navier-Stokes that fails global well-posedness for somewhat topological reasons.