This dissertation presents a search for dark matter particles produced in association with a Z boson in proton-proton collisions. The dataset consists of 139 fb^{-1} of collision events with centre-of-mass energy of 13 TeV, and was collected by the ATLAS detector from 2015-2018 at the Large Hadron Collider. Signal region events are required to contain a Z boson that decays leptonically to either e^+e^- or μ^+μ^-, and a significant amount of missing transverse momentum, which indicates the presence of undetected particles. Two types of dark matter models are studied: (1) simplified models with an s-channel axial-vector or vector mediator that couples to dark matter Dirac fermions, and (2) two-Higgs-doublet models with an additional pseudo-scalar that couples to dark matter Dirac fermions. The main Standard Model background sources are ZZ, WZ, non-resonant l^+l^-, and Z+jets processes, which are estimated using a combination of data and/or simulation. A new reweighting technique is developed for estimating the Z+jets background using γ+jets events in data; the resulting estimate significantly improves on the statistical and systematic errors compared to the estimate obtained from simulation. The observed data in the signal region are compared to Standard Model prediction using a transverse mass discriminant distribution. No significant excess in data is observed for the simplified models and two-Higgs-doublet models studied. A statistical analysis is performed and several exclusion limits are set on the parameters of the dark matter models. Results are compared to direct detection experiments, the CMS experiment, and other ATLAS searches. Prospects and improvements for future iterations of the search are also presented. / Graduate