Teleparallelism refers to any theory of gravitation based on the teleparallel framework.
Independently in 1967, Hayashi and Nakano revived Einstein's idea, and Pellegrini and Plebanski formulated the gauge theory of the spacetime translation group.
Its geometry is Weitzenböck's: manifolds that can be 'Weitzenböck flat' while carrying nonzero torsion instead of curvature.