The statistical properties of the strong coherent vortices observed in numerical simulations of isotropic turbulence are studied. When compiled at axial vorticity levels ω/ω′∼Re1/2λ, where ω′ is the r.m.s. vorticity magnitude for the flow as a whole, they have radii of the order of the Kolmogorov scale and internal velocity differences of the order of the r.m.s. velocity of the flow u′. Theoretical arguments are given to explain these scalings. It is shown that the filaments are inhomogeneous Burgers' vortices driven by an axial stretching which behaves like the strain fluctuations of the background flow. It is suggested that they are the strongest members in a class of coherent objects, the weakest of which have radii of the order of the Taylor microscale, and indirect evidence is presented that they are unstable. A model is proposed in which this instability leads to a cascade of coherent filaments whose radii are below the dissipative scale of the flow as a whole. A family of such cascades separates the self-similar inertial range from the dissipative limit. At the vorticity level given above, the filaments occupy a volume fraction which scales as Re−2λ, and their total length increases as O(Reλ). The length of individual filaments scales as the integral length of the flow, but there is a shorter internal length of the order of the Taylor microscale.