Whilst the Large Hadron Collider (LHC) has spent just over a decade pushing experimental boundaries in energy and intensity of proton-proton collisions, it has yet to discover new physics beyond the Standard Model. The Standard Model remains one of the most robustly tested theories of all time, despite evidence that it is incomplete, such as its inability to account for dark matter or the weakness of gravity. It is thus highly motivated that new particles should appear in proton-proton collisions at the ATLAS detector, around the TeV scale. This thesis contains three searches for such new particles which represent substantial improvements to constraints on possible new physics beyond the Standard Model. They demonstrate that data remains consistent with the Standard Model in collision events with two leptons in the final state.

The first two searches seek electroweak supersymmetry, using 80.5 fb⁻¹ and the full 139.5 fb⁻¹ of the ATLAS Run 2 dataset, respectively. The first search placed the first ATLAS Run 2 limits on chargino pair production decaying via W-bosons, excluding chargino masses up to 410 GeV for a massless neutralino. This was surpassed by the second search, resulting in world-leading exclusion limits on this process, up to 420 GeV, as well as excluding direct slepton production with slepton masses up to 700 GeV and chargino pair production decaying via sleptons or sneutrinos with chargino masses up to 1 TeV.

The third search uses an innovative technique to pursue signs of lepton charge-flavour asymmetry in the full Run 2 ATLAS dataset. Whilst it can be shown that the ratio of e⁺μ⁻ to e⁻μ⁺ events within Standard Model proton-proton collisions should not exceed one, this need not be the case for collision processes beyond the Standard Model. This search measures the ratio in data to be 0.988±0.005 and 0.987±0.005 in two regions of phase space (one with missing transverse momentum and one with a hard jet in the final state, respectively), as well as providing measurements binned in different variables. Furthermore, the search places the first direct exclusion limits at the LHC for single-production of a smuon within an R-parity violating supersymmetry model with the λ'₂₃₁ coupling switched on, excluding left-handed smuons up to 1.2 TeV for neutralino masses below the top quark mass and λ'₂₃₁ couplings up to one. It also interprets results in a scalar leptoquark model, placing the first LHC limits on a singly-produced leptoquark decaying to an eμ final state, excluding masses up to 2.2 TeV for leptoquark couplings λ(eu) and λ(µc) up to a value of one.