  We present the results of the computation of the next-to-leading order QCD
corrections to the production cross section of a Higgs boson in association
with a top-antitop pair at the LHC, including the three relevant dimension-six
operators ($O_{t \varphi }, O_{\varphi G}, O_{tG}$) of the standard model
effective field theory. These operators also contribute to the production of
Higgs bosons in loop-induced processes at the LHC, such as inclusive Higgs,
$Hj$ and $HH$ production, and modify the Higgs decay branching ratios for which
we also provide predictions. We perform a detailed study of the cross sections
and their uncertainties at the total as well as differential level and of the
structure of the effective field theory at NLO including renormalisation group
effects. Finally, we show how the combination of information coming from
measurements of these production processes will allow to constrain the three
operators at the current and future LHC runs. Our results lead to a significant
improvement of the accuracy and precision of the deviations expected from
higher-dimensional operators in the SM in both the top-quark and the
Higgs-boson sectors and provide a necessary ingredient for performing a global
EFT fit to the LHC data at NLO accuracy.
