  Absolute parallelism (AP) geometry is frequently used for physical
applications. Although it is wider than Riemannian geometry, it has two main
defects. The first is that its path equation does not represent physical
trajectories of any test particle. The second is the identical vanishing of its
curvature tensor. The present work shows that parameterizing this geometry
would solve the two problems. Furthermore, the resulting parameterized
AP-structure is more general than both the conventional AP-structure and the
Riemannian structure. Also, it is shown that it can be reduced to one or the
other, of these two geometric structures, in some special cases. The structure
obtained is more appropriate for physical applications, especially in
constructing field theories gauging gravity.
