A probability weighting function w(p) is a prominent feature of several non-expected utility theories, including prospect theory and rank-dependent models. Empirical estimates indicate that w(p) is regressive (first w(p) > p, then w(p) < p), s-shaped (first concave, then convex), and asymmetrical (intersecting the diagonal at about 1/3). The paper states axioms for several w(p) forms, including the compound invariant, w(p) = exp{-{-ln p}$^{\alpha}$}, 0 < $\alpha$ < 1, which is regressive, s-shaped, and with an invariant fixed point and inflection point at 1/e = .37.