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The authors discuss the existence of fractional quantised Hall states in other than the lowest (n= 0) Landau level. They present the results of numerical studies of systems with up to nine particles in the n= 1 Landau level. They find that a fractional quantised Hall state is likely for filling fractions of the n= 1 Landau level, nu 1, with nu 1-1= 5. At nu 1-1= 3 they find that the Laughlin trial wavefunctions is not a good candidate ground-state wavefunction. If there is a gap at nu 1-1 it is predicted to be small. At nu 1-1= 7/2 they predict the existence of a fractional quantised Hall state despite the small or non-existent gap at nu 1-1= 3. They also show that introducing truncated pseudopotential interactions is not always a physical procedure outside of the lowest Landau level.