In this thesis, searches for magnetic monopoles and highly ionising particles carried out at the ATLAS and MoEDAL experiments at CERN, is presented. The searches were performed using 7 fb-1 and 0.75 fb-1 of proton-proton collision data collected with the ATLAS and MoEDAL experiments respectively, both at the centre-of-mass energy of 8 TeV. The thesis describes two complimentary techniques to search for magnetic monopoles in the two experiments. In ATLAS, a novel trigger is developed to select events with monopole-like signatures and subsequent selections are applied on the ionisation and width of the calorimeter clusters of such particles. No event is found in the signal region, leading to production cross-section upper limits in the mass range 200--2500 GeV for magnetic monopoles with magnetic charge in the range 0.5gD<|g|<2.0gD, where gD is the Dirac charge, and for stable particles with electric charge in the range 10<|z|<60. Model-dependent limits are presented in given pair-production scenarios, and model-independent limits are presented in fiducial regions of particle energy and pseudorapidity. In MoEDAL, a prototype trapping detector deployed around the LHCb interaction point. These samples are monitored offline for the presence of trapped magnetic charge at a remote superconducting magnetometer facility. No magnetic charge exceeding 0.5gD (where gD is the Dirac magnetic charge) is measured in any of the exposed samples, allowing limits to be placed on monopole production in the mass range 100 GeV≤m≤ 3500 GeV. Model-independent cross-section limits are presented in fiducial regions of monopole energy and direction for 1gD≤|g|≤6gD, and model-dependent cross-section limits are obtained for Drell-Yan pair production of spin-1/2 and spin-0 monopoles for 1gD≤|g|≤4gD. Under the assumption of Drell-Yan cross sections, mass limits are derived for |g|=2gD and |g|=3gD for the first time at the LHC, surpassing the results from previous collider experiments.