  We observe that the pp wave limit of $AdS_5\times M^5$ compactifications of
type IIB string theory is universal, and maximally supersymmetric, as long as
$M^5$ is smooth and preserves some supersymmetry. We investigate a specific
case, $M^5=T^{1,1}$. The dual ${\cal N}=1$ SCFT, describing D3-branes at a
conifold singularity, has operators that we identify with the oscillators of
the light-cone string in the universal pp-wave background. The correspondence
is remarkable in that it relies on the exact spectrum of anomalous dimensions
in this CFT, along with the existence of certain exceptional series of
operators whose dimensions are protected only in the limit of large `t Hooft
coupling. We also briefly examine the singular case $M^5=S^5/Z_2$, for which
the pp wave background becomes a $Z_2$ orbifold of the maximally supersymmetric
background by reflection of 4 transverse coordinates. We find operators in the
corresponding ${\cal N}=2$ SCFT with the right properties to describe both the
untwisted and the twisted sectors of the closed string.
