  The production of a top-antitop quark pair in association with a $W$ boson
($t\bar tW$) is one of the heaviest signatures currently probed at the Large
Hadron Collider (LHC). Since the first observation reported in 2015 the
corresponding rates have been found to be consistently higher than the Standard
Model predictions, which are based on next-to-leading order~(NLO) calculations
in the QCD and electroweak (EW) interactions. We present the first
next-to-next-to-leading order (NNLO) QCD computation of $t\bar tW$ production
at hadron colliders. The calculation is exact, except for the finite part of
the two-loop virtual corrections, which is estimated using two different
approaches that lead to consistent results within their uncertainties. We
combine the newly computed NNLO QCD corrections with the complete NLO QCD+EW
results, thus obtaining the most advanced perturbative prediction available to
date for the \ttW inclusive cross section. The tension with the latest ATLAS
and CMS results remains at the $1\sigma-2\sigma$ level.
