  We solve the local equivalence problem for second order (smooth or analytic)
ordinary differential equations. We do so by presenting a {\em complete
convergent normal form} for this class of ODEs. The normal form is optimal in
the sense that it is defined up to the automorphism group of the model (flat)
ODE $y"=0$. For a generic ODE, we also provide a unique normal form. By doing
so, we give a solution to a problem which remained unsolved since the work of
Arnold. The method can be immediately applied to important classes of second
order ODEs, in particular, the Painlev\'e equations.
  As another application of the convergent normal form, we discover
distinguished curves associated with a differential equation that we call {\em
chains}.
