This thesis presents a search for long-lived particles decaying into displaced electrons and/or muons with large impact parameters. This signature provides unique sensitivity to the production of theoretical lepton-partners, sleptons. These particles are a feature of supersymmetric theories, which seek to address unanswered questions in nature. The signature searched for in this thesis is difficult to identify, and in fact, this is the first time it has been probed at the Large Hadron Collider (LHC). It covers a long-standing gap in coverage of possible new physics signatures. This thesis describes the special reconstruction and identification algorithms used to select leptons with large impact parameters and the details of the background estimation. The results are consistent with background, so limits on slepton masses and lifetimes in this model are calculated at 95% CL, drastically improving on the previous best limits from the Large Electron Positron Collider (LEP).