  In this paper we study Chow motives whose identity map is killed by a natural
number. Examples of such objects were constructed by Gorchinskiy-Orlov. We
introduce various invariants of torsion motives, in particular, the $p$-level.
We show that this invariant bounds from below the dimension of the variety a
torsion motive $M$ is a direct summand of and imposes restrictions on motivic
and singular cohomology of $M$. We study in more details the $p$-torsion
motives of surfaces, in particular, the Godeaux torsion motive. We show that
such motives are in 1-to-1 correspondence with certain Rost cycle submodules of
free modules over $H^*_{et}$. This description is parallel to that of mod-$p$
reduced motives of curves.
