  We exploit fluctuational electrodynamics to present trace expressions for the
torque experienced by arbitrary objects in a passive, non-absorbing,
rotationally invariant background environment. Specializing to a single object,
this formalism, together with recently developed techniques for calculating
bounds via Lagrange duality, is then used to derive limits on the maximum
Casimir torque that a single object with an isotropic electric susceptibility
can experience when out of equilibrium with its surrounding environment. The
maximum torque achievable at any wavelength is shown to scale in proportion to
body volumes in both subwavelength (quasistatics) and macroscopic (ray optics)
settings, and come within an order of magnitude of achievable torques on
topology optimized bodies. Finally, we discuss how to extend the formalism to
multiple bodies, deriving expressions for the torque experienced by two
subwavelength particles in proximity to one another.
