  We report on the calculation of the next-lo-leading order QCD corrections to
Higgs boson production and decay in association with top quarks. We consider
leptonic decays of top quarks leading to the hadronic process $pp\to e^+\nu_e\,
\mu^-\bar{\nu}_\mu \, b\bar{b}\,H\, (H\to X)$ at the LHC with $\sqrt{s}=13$
TeV. All resonant as well as non-resonant Feynman diagrams, interferences and
off-shell effects are included for the top quark and $W$ gauge boson. Decays of
the Higgs boson, on the other hand, are included in the
narrow-width-approximation. Specifically, we consider Higgs boson decays into
$b\bar{b}$, $\tau^+\tau^-$, $\gamma\gamma$ and $e^+e^-e^+e^-$. Numerical
results are given at the integrated and differential fiducial level for various
factorisation and renormalisation scale choices and different PDF sets. We
study the main theoretical uncertainties that are associated with neglected
higher order terms in the perturbative expansion and with different
parametrisations of the PDFs. Furthermore, we examine the size of the off-shell
effects by an explicit comparison to the calculation in the full
narrow-width-approximation. Finally, the impact of the contributions induced by
the bottom-quark parton density is investigated.
