  Working in the framework of nilpotent geometry, we give a unified scheme for
the equivalence problem of geometric structures which extends and integrates
the earlier works by Cartan, Singer-Sternberg, Tanaka, and Morimoto. By giving
a new formulation of the higher order geometric structures and the universal
frame bundles, we reconstruct the step prolongation of Singer-Sternberg and
Tanaka. We then investigate the structure function $\gamma$ of the complete
step prolongation of a normal geometric structure by expanding it into
components $\gamma = \kappa + \tau + \sigma$ and establish the fundamental
identities for $\kappa$, $\tau$, $\sigma$. This then enables us to study the
equivalence problem of geometric structures in full generality and to extend
applications largely to the geometric structures which have not necessarily
Cartan connections.
  Among all we give an algorithm to construct a complete system of invariants
for any higher order normal geometric structure of constant symbol by making
use of generalized Spencer cohomology group associated to the symbol of the
geometric structure. We then discuss thoroughly the equivalence problem for
geometric structure in both cases of infinite and finite type. We also give a
characterization of the Cartan connections by means of the structure function
$\tau$ and make clear where the Cartan connections are placed in the
perspective of the step prolongations.
