In total 1317 di-lepton events with significant missing transverse momentum were identified in a total data sample of 680 pb^-1 collected at e+e- centre-of-mass energies ranging from 183 GeV to 209 GeV. The number of di-lepton events, the dependence on centre-of-mass energy, and the event properties are consistent with expectations from Standard Model processes, predominantly W+W- production with both W bosons decaying leptonically. This topology is also an experimental signature for the pair production of new particles that decay to a charged lepton accompanied by one or more invisible particles. No evidence for new phenomena is apparent. Upper limits are presented on the production cross-section multiplied by the relevant branching ratio squared for sleptons, leptonically decaying charginos and charged Higgs bosons. Mass limits are also given.