CERN-LHC. A search for supersymmetry (SUSY) in events with large missing transverse momentum, jets, at least one hadronically decaying tau lepton and zero or one additional light leptons (electron/muon), has been performed using 20.3 fb^-1 of proton-proton collision data at sqrt(s)=8 TeV recorded with the ATLAS detector at the Large Hadron Collider. No excess above the Standard Model background expectation is observed in the various signal regions and 95% confidence level upper limits on the visible cross section for new phenomena are set. The results of the analysis are interpreted in several SUSY scenarios, significantly extending previous limits obtained in the same final states. In the framework of minimal gauge-mediated SUSY breaking models, values of the SUSY breaking scale Lambda below 63 TeV are excluded, independently of tan(beta). Exclusion limits are also derived for an mSUGRA/CMSSM model, in both the R-parity-conserving and R-parity-violating case. A further interpretation is presented in a framework of natural gauge mediation, in which the gluino is assumed to be the only light coloured sparticle and gluino masses below 1090 GeV are excluded.