  A geometric structure (FAP-structure), having both absolute parallelism and
Finsler properties, is constructed. The building blocks of this structures are
assumed to be functions of position and direction. A non-linear connection
emerges naturally and is defined in terms of the building blocks of the
structure. Two linear connections, one of Berwald type and the other of the
Cartan type, are defined using the non-linear connection of the FAP. Both
linear connections are non-symmetric and consequently admit torsion. A metric
tensor is defined in terms of the building blocks of the structure. The
condition for this metric to be a Finslerian one is obtained. Also, the
condition for an FAP-space to be an AP-one is given.
