  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 examples, i.e. $S^6$, $\mathbb{CP}^3$, the Wallach space $SU(3)/T^2$
and the biquotient $SU(3)//T^2$. We also classify, up to equivariant
diffeomorphism, certain actions without exceptional orbits and show that there
are strong restrictions on the exceptional strata.
