This dissertation presents a search for single-top quarks produced in association with missing energy, denoted as monotop, using 20.3~fb$^{\mathhyphen 1}$ of proton-proton collision data at $\sqrt{s}=8$~TeV recorded with the ATLAS detector. The search is conducted in two separate analyses consisting of a cut-based and a multivariate approach. As no deviation from the Standard Model prediction is observed, upper limits are set on the monotop production cross-section times branching ratio for resonant and non-resonant production of an invisible exotic state in association with a single-top quark. In the case of resonant production, for a spin-0 resonance with a mass of 500~GeV, an effective coupling strength above 0.13 is excluded at the 95\% confidence level for an invisible spin-1/2 state with mass up to 100~GeV using the cut-based approach. In the case of non-resonant production, an effective coupling strength above 0.2 is excluded at the 95\% confidence level for an invisible spin-1 state with mass up to 657~GeV and 799~GeV using the cut-based and multivariate approaches, respectively. In addition to setting observed limits at $\sqrt{s}=8$~TeV, expected upper limits are set on the non-resonant production model for future levels of integrated luminosity expected to be recorded at $\sqrt{s}=13$~TeV.
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