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We present new NLO sets of parton distributions in real photons based on ascheme invariant definition of the non-perturbative input. We compare thetheoretical predictions with LEP data and a best fit allows us to constrain theparameters of the distributions. The shape of the gluon distribution is poorlyconstrained and we consider the possibility to measure it in photoproductionexperiments. Three parametrizations which aim to take into account thescattering of LEP data are proposed. They are compared to other NLOparametrizations.Comment: 38 pages, 23 Postscript figures, fig. 6,7,8,9 improved, comparisons  with other NLO parametrizations, added reference

New NLO parameterizations of the parton distributionsin real photons