Searches for new physics with leptons and invisible particles at the ATLAS experiment at the LHC

The Standard Model of Particle Physics (SM), while successful at describing almost all subatomic phenomena observed to date, has some glaring open questions: the Higgs boson mass is unstable at high energies, there is no dark matter candidate, and the prediction and measurement of $(g − 2)_\mu$ are in tension. Supersymmetric (SUSY) extensions to the Standard Model could provide possible answers to all these problems, and some SUSY particles could be produced at the Large Hadron Collider. This thesis describes three areas of work aimed at finding evidence for SUSY at the ATLAS Experiment of the Large Hadron Collider. First, the performance and development of the ATLAS $E_T^\text{miss}$ trigger is discussed. Many theorised SUSY process are predicted to produce events with $E_T^\text{miss}$, a large imbalance in the momentum sum of detectable particles, as SUSY particles could escape detection. The $E_T^\text{miss}$ trigger aims to select such events during data-taking to be saved to disk for analysis. Next, a search for new physics in events with four or more leptons is described. This search is optimised for certain SUSY models, but is also sensitive to a wide range of beyond-the-Standard- Model (BSM) processes. No significant deviation from the SM is observed. From this observation, limits are set on general gauge-mediated SUSY models, $R$-parity violating SUSY models, and BSM contributions. Finally, a machine-learning-based strategy is proposed to probe an experimentally challenging but phenomenologically favoured range of SUSY parameters in events with two leptons and $E_T^\text{miss}$. The strategy is found to not only be competitive with a more traditional cuts-based strategy, but it also provides the event yields necessary for the measurement of SUSY particle properties.