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The teleparallelism theory of gravity is presented as a constrained Poincark gauge theory. Arguments are given in favor of a two-parameter family of field Lagrangians quadratic in torsion. The inclusion of a "parity violating" term in the field Lagrangian avoids difficulties with the initial-data problem, recently discussed in the literature. Several new exact solutions of the corresponding teleparallelism theory give considerable insight into its physical consequences. The resulting general field equations are analyzed in the weak-field approximation excluding ghosts and tachyons. The physical meaning of the six additional components of the tetrad field (as compared with the metric) appears naturally from our theory and is made clear.

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