  A measurement of the production cross section for a top quark pair in
association with a $Z$ boson ($t\bar{t}Z$) is presented in this PhD thesis.
Final states with exactly three charged leptons (electrons or muons) are used,
taking into account the decay of the top quark pair in the lepton+jets channel
and the decay of the $Z$ boson into two charged leptons. The dataset used for
this analysis corresponds to 36.1 fb$^{-1}$ of proton-proton collisions at a
centre-of-mass energy of 13 TeV, recorded during 2015 and 2016 by the ATLAS
detector at the Large Hadron Collider. The result of a profile likelihood fit
to the event yields in four signal enriched regions and two background enriched
regions is
$\sigma_{t\bar{t}Z}=966^{+114}_{-102}(\text{stat.})^{+115}_{-114}(\text{syst.})$
fb. The observed (expected) significance is $7.2$ ($6.4$) standard deviations
from the background-only hypothesis. Within the experimental uncertainties, the
result is in good agreement with the Standard Model prediction. This result is
compared with two other $t\bar{t}Z$ analysis channels, using the same dataset
but different lepton multiplicities. The analysis presented here is found to be
the most sensitive one in terms of observed significance. The result of a
combined fit of all three analysis channels is discussed. Two feasibility
studies of possible future $t\bar{t}Z$ analysis techniques are demonstrated.
