This thesis presents a search for exotic, neutral, long-lived particles, which decay to final states that include a pair of electrons or a pair of muons. The search is performed with proton-proton collision data collected by the CMS detector at the LHC at a centre-of-mass energy √s = 8 TeV. The experimental signature consists of a pair of charged leptons originating from a vertex that is significantly displaced from the centre of the CMS detector. This is a very striking signature and would clearly indicate the presence of new physics beyond the Standard Model if observed. No significant excess of events was observed above the Standard Model expectations. Therefore, upper limits are placed on the production of long-lived particles in the context of two benchmark models, as a function of the lifetime of the long-lived particle.