Quantum gravity and time reversibility

The meaning of time-reversal and $\mathrm{CPT}$ invariances of a theory is discussed both in the context of theories defined on flat spacetime as well as in general relativity. It is argued that quantum gravity cannot be time-reversal or $\mathrm{CPT}$ invariant; that an "arrow of time" must be fundamentally built into the theory. However, a weaker form of $\mathrm{CPT}$ invariance could still hold, in which case the fundamental "arrow of time" would not show up in the measurements of observers who perform scattering experiments. Consequences of this weaker hypothesis are explored.