In this thesis, we discuss the development and use cases of the public software CheckMATE which is designed to allow for easy tests of theories beyond the Standard Model against current results from the Large Hadron Collider (LHC). We illustrate the general functionality of this tool and provide hands-on examples to explain how it can be used to test results from the ATLAS and CMS experiments. In addition, we explain how new analyses can be conveniently added to the existing framework. This tool is then used to project a search for monojet final states to a high luminosity LHC with a centre-of-mass energy of 14 TeV. Here, our prospective analysis is used to determine the expected sensitivity to a Higgs Portal scenario which couples the Standard Model to a hidden sector via an invisibly decaying second heavy scalar. We show that complementary bounds to those derived from Higgs boson searches in 8 TeV LHC data can be set, however only if a significant reduction of the current systematic uncertainties for the background estimates of such a search can be achieved. Furthermore, we use CheckMATE and its large set of implemented searches for natural Supersymmetry to show how an extension of the Minimal Supersymmetric Standard Model by an additional chiral gauge singlet typically reduces the LHC sensitivity. In the context of R-parity violating Supersymmetry, we go beyond CheckMATE and the LHC and derive how the expected sensitivity of the proposed fixed-target experiment SHiP to observe long-lived neutralinos produced via rare Standard Model meson decays can significantly improve existing bounds from low energy observations
