In my experience of GR the torsion tensor is typically taken to vanish, but I believe this is more or less for simplicity than anything else.
What does the torsion tensor do? What physical effects would it manifest? Do it's components get nice explanations? [For example, the $(i,j)^{th}$ component of the Stress-Energy tensor is the flux of $i$-momentum in the $j$-direction.]
