This talk will survey some of the recent work (with Shkoller, Ratiu, and Holm) on the averaged Euler equations. We will begin with a sketch of the derivation of the equations. Then we show how the equations may be regarded as geodesic equations for the $H^1$ metric on the volume preserving diffeomorphism group, as Arnold did with the $L^2$ metric for the Euler equations. Then we will present some of the analytical theorems, as in Ebin and Marsden, including the convergence as viscosity tends to zero, even in the presense of boundaries. We will also briefly indicate some of the interesting computational aspects of the equations, including high Reynolds number computations and vortex merger without dissipation.