The overwhelming astrophysical evidence for Dark Matter is an important motivation to search for new physics at the Large Hadron Collider (LHC) at CERN. While the Standard Model of particle physics is able to predict measurements and observations to an astounding precision, it does not provide a candidate particle for Dark Matter. If possibly produced in high-energy proton-proton collisions, such particles would traverse the detectors without leaving a signal. Hence, searches rely on the resulting momentum imbalance in the transverse plane. One particular extension of the Standard Model that allows for a Dark Matter candidate is Supersymmetry. Since the supersymmetric partner of the top quark is expected to be relatively light it could be in reach of LHC experiments and possibly detected. This thesis presents a study of the validity of commonly-used effective field theory models of Dark Matter production at the LHC. It shows that in a significant fraction of events the assumptions of an effective field theory description are not justified, which requires a redefinition of strategy when interpreting LHC results in terms of Dark Matter production. The results from a search for new phenomena in events with an energetic jet and large missing transverse energy is presented. It is performed on 20.3 fb^-1 of 8 TeV pp collision data, recorded by the ATLAS detector at the LHC. No evidence for new physics was observed. The results are interpreted in terms of Dark Matter production within an effective model as well as using a Simplified Model, motivated by the findings of the validity study. Subsequently, this result and two other ATLAS searches are studied in a detailed reinterpretation in terms of Simplified Models of Dark Matter production. A large range of parameters is tested for three different Simplified Models. The study revealed, that a dedicated optimisation in view of Simplified Models would be beneficial, especially in the regime of small missing transverse energy. Final states of Dark Matter and top quarks are well-motivated by models with a scalar or pseudo-scalar particle mediating the interaction between Standard Model and Dark Matter particles. The resulting final state is similar to that of the production of supersymmetric top partners. A search for new phenomena in such final states of top quark pairs and large missing transverse energy, performed on 13.2 fb^-1 of data from 13 TeV pp collisions is presented. An excess of data events over the Standard Model background prediction of 3.3 sigma was observed in a signal region optimised for Dark Matter signals. Interpretations of the results are presented for two decay scenarios of supersymmetric top quark partners and for Dark Matter production in association with top quarks.