Differential Invariants, Double Fibrations and Painlevé Equations

Our aims in this paper are to review two complementary approaches to the geometry of second order o.d.e.s $$y'' = F\left( {x,y,y'} \right)$$ (0.1) , to elucidate the correspondence between these approaches and finally to present some applications to equations (0.1) having the Painleve property.