  We investigate modified theories of gravity in the context of teleparallel
geometries with possible Gauss-Bonnet contributions. The possible coupling of
gravity with the trace of the energy-momentum tensor is also taken into
account. This is motivated by the various different theories formulated in the
teleparallel approach and the metric approach without discussing the exact
relationship between them. Our formulation clarifies the connections between
different well known theories. For instance, we are able to formulate the
correct teleparallel equivalent of Gauss-Bonnet modified general relativity,
amongst other results. Finally, we are able to identify modified gravity models
which have not been studied in the past. These appear naturally within our
setup and would make a interesting starting point for further studies.
