  The present paper concerns the dynamics of surface diffeomorphisms. Given a
diffeomorphism $f$ of a surface $S$, the \emph{torsion} of the orbit of a point
$z\in S$ is, roughly speaking, the average speed of rotation of the tangent
vectors under the action of the derivative of $f$, along the orbit of $z$ under
$f$. The purpose of the paper is to identify some situations where there exist
measures and orbits with non-zero torsion. We prove that every area preserving
diffeomorphism of the disc which coincides with the identity near the boundary
has an orbit with non-zero torsion. We also prove that a diffeomorphism of the
torus $\mathbb{T}^2$, isotopic to the identity, whose rotation set has
non-empty interior, has an orbit with non-zero torsion.
