Astronomical and cosmological observations support the existence of invisible matter that can only be detected through its gravitational effects, thus making it very difficult to study. This component, called dark matter, makes up about 26.8\% of the known universe. Experiments at the LHC, located at CERN, search for new particles to be dark matter candidates. A dark matter production can consist of an excess of events with a single final-state object X recoiling against large amount of missing momentum of energy called mono-X signal. The studies presented in this thesis are focused on the mono-X signature, with X being a top quark, named mono-top. The topology is studied, where the $W$ boson from the associated top-quark decays into a lepton (electron or a muon) and a neutrino. Firstly, a sensitivity search of dark matter production in an extension of the Standard Model featuring a two-Higgs-doublet model and an additional pseudo-scalar is presented. The pseudo-scalar is the mediator which decays to the dark matter particles. This analysis uses all the data collected by the ATLAS experiment at the LHC during Run 2 (2015--2018) corresponding to an integrated luminosity of 139 fb$^{-1}$, at a centre-of-mass energy of 13 TeV. A multivariate analysis based on a boosted decision tree is performed in order to enhance the discrimination of signal events from the main background. The results are expressed as 95% confidence level limits on the parameters of the signal models considered. No significant excess is found with respect to Standard Model predictions. The region below of $m_{H^{\pm}}$ = 800 GeV (1100 GeV) with $\tan\beta$ = 0.3 is excluded with 95% confidence level of the observed (expected) limit. Furthermore, the prospect of the potential discovery of the non-resonant production of an exotic state decaying into a pair of dark matter candidates in association with a right-handed top quark in the context of an effective dark matter flavour-changing neutral interaction, at the HL-LHC is presented as well. The HL-LHC project is expected to operate at a centre-of-mass energy of 14 TeV aiming to provide a total integrated luminosity of 3000 fb$^{-1}$. The number of signal and background events is estimated from simulated particle-level truth information after applying smearing functions to mimic an upgraded ATLAS detector response in the HL-LHC environment. The expected exclusion limit (discovery reach) at 95% confidence level for the mass of the exotic state is calculated to be 4.6 TeV (4.0 TeV), using a multivariate analysis based on a boosted decision tree.