We address the question whether the lightest scalar mesons sigma and kappa are tetraquarks, as is strongly supported by many phenomenological studies. We present a search for possible light tetraquark states with J^PC=0^++ and I=0, 1/2, 3/2, 2 on the lattice. The spectrum is determined using the generalized eigenvalue method with a number of tetraquark interpolators at the source and the sink. In all the channels, we unavoidably find lowest scattering states pi(k)pi(-k) or K(k)pi(-k) with back-to-back momentum k=0,2*pi/L,.. . However, we find an additional light state in the I=0 and I=1/2 channels, which may be related to the observed resonances sigma and kappa with a strong tetraquark component. In the exotic repulsive channels I=2 and I=3/2, where no resonance is observed, we find no light state in addition to the scattering states.
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