In 1864, James Clerk Maxwell showed that a system of N spherical particles in d-dimensions is mechanically stable only if the number, z, of two-point contacts between particles exceeds zc = 2d. Systems with z=zc are isostatic. Recent work confirms that randomly packed spheres are isostatic at the point J where the volume fraction φ reaches the critical value φc necessary to support shear and that the mechanics of this isostatic state determine behavior at volume fractions above φc. The square and kagome lattice with nearest neighbor springs are isostatic. This talk will discuss the mechanical properties and phonon spectrum of nearly isostatic versions of these lattices in which next-nearest-neighbor springs with a variable spring constant are added either homogeneously or randomly. In particular, it will show that these lattices exhibit what appears to be universal features of nearly isostatic systems, namely characteristic lengths that diverge as 1/(z-zc) and frequencies that vanish as (z-zc). The shear elastic modulus depends on the geometry of the isostatic network and is not universal. Response near z=zc in the random case is highly nonaffine.