  Naturalness bounds on weak scale supersymmetry in the context of radiative
breaking of the electro-weak symmetry are analyzed. In the case of minimal
supergravity it is found that for low tan\beta and for low values of fine
tuning Phi, where Phi is defined essentially by the ratio \mu^2/M_Z^2 where \mu
is the Higgs mixing parameter and M_Z is the Z boson mass, the allowed values
of the universal scalar parameter m_0, and the universal gaugino mass m_{1/2}
lie on the surface of an ellipsoid with radii fixed by \Phi leading to tightly
constrained upper bounds \sim \sqrt\Phi.Thus for tan\beta\leq 2(\leq 5) it is
found that the upper limits for the entire set of sparticle masses lie in the
range < 700 GeV (<1.5TeV) for any reasonable range of fine tuning (\Phi\leq
20). However, it is found that there exist regions of the parameter space where
the fine tuning does not tightly constrain m_0 and m_{1/2}. Effects of non-
universalities in the Higgs sector and in the third generation sector on
naturalness bounds are also analyzed and it is found that non-universalities
can significantly affect the upper bounds. It is also found that achieving the
maximum Higgs mass allowed in supergravity unified models requires a high
degree of fine tuning. Thus a heavy sparticle spectrum is indicated if the
Higgs mass exceeds 120 GeV. The prospect for the discovery of supersymmetry at
the Tevatron and at the LHC in view of these results is discussed.
