We study the topology of closed, simply-connected, 6-dimensional Riemannian manifolds of positive sectional curvature which admit isometric actions by SU(2) or SO(3). We show that their Euler characteristic agrees with that of the known exam- ples, i.e. S6, CP3, the Wallach space SU(3)=T 2 and the biquotient SU(3)==T 2. We also classify, up to equivariant dieomorphism, certain actions without exceptional orbits and show that there are strong restrictions on the exceptional strata.