  We investigate density perturbations sourced by a curvaton with a generic
energy potential. The key feature of a curvaton potential which deviates from a
quadratic is that the curvaton experiences a non-uniform onset of its
oscillation. This sources additional contributions to the resulting density
perturbations, and we especially find that the non-Gaussianity parameter f_NL
can become large with either sign no matter whether the curvaton dominates or
subdominates the universe when it decays. Such non-quadratic curvaton
potentials are required in order to produce a red-tilted density perturbation
spectrum (without invoking large-field inflation), and are also motivated by
explicit curvaton models based on microscopic physics. We further apply our
generic results to the case where the curvaton is a pseudo-Nambu-Goldstone (NG)
boson of a broken U(1) symmetry, and show that the resulting density
perturbations are strongly enhanced towards the hilltop region of the energy
potential, accompanied by a mild increase of the non-Gaussianity. Such hilltop
NG curvatons can produce observationally suggested density perturbations under
wide ranges of inflation/reheating scales, and further predict the
non-Gaussianity of the density perturbations to lie within the range 10 <~ f_NL
<~ 30.
