  We compute the total cross section for $t\bar{t}t\bar{t}$ production at
next-to-leading logarithmic (NLL$^{\prime}$) accuracy. This is the first time
resummation is performed for a hadron-collider process with four colored
particles in the final state. The calculation is matched to the next-to-leading
order strong and electroweak corrections. The NLL$^{\prime}$ corrections
enhance the total production rate by 15\%. The size of the theoretical error
due to scale variation is reduced by more than a factor of two, bringing the
theoretical error significantly below the current experimental uncertainty of
the measurement.
