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The Pfaffian quantum Hall states, which can be viewed as involving pairingeither of spin-polarized electrons or of composite fermions, are generalized byfinding the exact ground states of certain Hamiltonians with k+1-bodyinteractions, for all integers k > 0. The remarkably simple wavefunctions ofthese states involve clusters of k particles, and are related to correlators ofparafermion currents in two-dimensional conformal field theory. The k=2 case isthe Pfaffian. For k > 1, the quasiparticle excitations of these systems areexpected to possess nonabelian statistics, like those of the Pfaffian. For k=3,these ground states have large overlaps with the ground states of the (2-body)Coulomb-interaction Hamiltonian for electrons in the first excited Landau levelat total filling factors \nu=2+3/5, 2+2/5.Comment: 11 pages Revtex in two column format with 4 eps figures included in  the M

Beyond paired quantum Hall states: Parafermions and incompressible states in the first excited Landau level