We study superfluorescence (SF) from spherical and cigar-shaped clouds of laser-cooled Rubidium atoms from the
5D5/2 level through the
6P3/2 level to the
5S1/2 ground level. The atomic system is initially excited to the
5D5/2 level from the ground state via two-photon excitation through the intermediate
5P3/2 level. The fluorescence on the $6P\text{\ensuremath{-}}5S$ transition at 420 nm is recorded using time-resolved measurements. The time delays of the observed SF emission peaks typically scale as $\ensuremath{\sim}{N}^{\ensuremath{-}1}$, where
N is the atom number, and are much smaller than the time delay expected for uncorrelated cascade fluorescence. Since
N is significantly smaller than the threshold number for SF on the 420 nm transition, and larger than the threshold number for the $5D\text{\ensuremath{-}}6P$ transition at $5.2\text{ }\ensuremath{\mu}\text{m}$, our observations suggest that the 420 nm SF emission is triggered by rapid deexcitation of the
5D to the
6P level via SF at $5.2\text{ }\ensuremath{\mu}\text{m}$. The observed SF time delays for 420 nm emission agree with SF time-delay estimates for the $5.2\text{ }\ensuremath{\mu}\text{m}$ transition. For spherical clouds, the SF is isotropic. For cigar-shaped clouds, the SF is highly anisotropic. Along the long axis of cigar-shaped atom clouds, SF and incoherent cascade fluorescence produce temporally well-resolved peaks in the detected signal. In this case, the SF component of the signal is highly concentrated along a direction in between the directions of the two almost parallel excitation beams. The observed SF intensities scale as
N, suggesting that the
5D level is regeneratively pumped during the SF decay.