Abstract:
Properties and experimental predictions of a broad class of supergravity grand unified models possessing an $SU(5)$-type proton decay and $R$ parity are described. Models of this type can be described in terms of four parameters at the Gut scale in addition to those of the Standard Model i.e. $m_o$ (universal scalar mass), $m_{1/2}$ (universal gaugino mass), $A_o$ (cubic soft breaking parameter) and $\tan\beta=/$. Thus the 32 SUSY masses can be expressed in terms of $m_o, m_{1/2}, A_o \tan\beta$ and the as yet unknown t-quark mass $m_t$. Gut thresholds are examined and a simple model leads to grand unification consistent with $p$-decay data when $0.114<\alpha_3 (M_z)<0.135$, in agreement with current values of $\alpha_3 (M_Z)$.Proton decay is examined for the superheavy Higgs triplet mass $M_{H_3}<10M_G(M_G\simeq 1.5 \times10^{16}$~GeV) and squarks and gluinos lighter than 1 TeV. Throughout most of the parameter space chargino-neutralino scaling relations are predicted to hold: $2m_{\tilde{Z}_1}\cong m_{\tilde{W}_1}\cong m_{\tilde{Z}_2}, m_{\tilde{W}_1}\simeq(1/4)m_{\tilde{g}}$ (for $\mu>0$) or $m_{\tilde{W}_1} \simeq(1/3)m_{\tilde{g}}$ (for $\mu<0$), while $m_{\tilde{W}_2}\cong m_{\tilde{Z}_3}\cong m_{\tilde{Z}_4}>>m_{\tilde{Z}_1}$. Future proton decay experiments combined with LEP2 lead to further predictions, e.g. for the entire parameter space either proton decay should be seen at these or the$\tilde{W}_1$ seen at LEP2. Relic density constraints on the $\tilde{Z}_1$ further constrain the parameter space e.g. so that $m_t<165$~GeV, $m_h<105$~GeV, $m_{\tilde{W}_1} <100$~GeV and $m_{\tilde{Z}_1}<50$~GeV when $M_{H_3}/M_G < 6$. (Invited talk at Les Rencontres de Physique de la Vallee D'Aoste}

Abstract:
A survey is given of supersymmetry and supergravity and their phenomenology. Some of the topics discussed are the basic ideas of global supersymmetry, the minimal supersymmetric Standard Model (MSSM) and its phenomenology, the basic ideas of local supersymmetry (supergravity), grand unification, supersymmetry breaking in supergravity grand unified models, radiative breaking of $SU(2) \times U(1)$, proton decay, cosmological constraints, and predictions of supergravity grand unified models. While the number of detailed derivations are necessarily limited, a sufficient number of results are given so that a reader can get a working knowledge of this field.

Abstract:
We present the results of an extensive exploration of the five-dimensional parameter space of the minimal $SU(5)$ supergravity model, including the constraints of a long enough proton lifetime ($\tau_p>1\times10^{32}\y$) and a small enough neutralino cosmological relic density ($\Omega_\chi h^2_0\le1$). We find that the combined effect of these two constraints is quite severe, although still leaving a small region of parameter space with $m_{\tilde g,\tilde q}<1\TeV$. The allowed values of the proton lifetime extend up to $\tau_p\approx1\times10^{33}\y$ and should be fully explored by the SuperKamiokande experiment. The proton lifetime cut also entails the following mass correlations and bounds: $m_h\lsim100\GeV$, $m_\chi\approx{1\over2}m_{\chi^0_2}\approx0.15\gluino$, $m_{\chi^0_2}\approx m_{\chi^+_1}$, and $m_\chi<85\,(115)\GeV$, $m_{\chi^0_2,\chi^+_1}<165\,(225)\GeV$ for $\alpha_3=0.113\,(0.120)$. Finally, the {\it combined} proton decay and cosmology constraints predict that if $m_h\gsim75\,(80)\GeV$ then $m_{\chi^+_1}\lsim90\,(110)\GeV$ for $\alpha_3=0.113\,(0.120)$. Thus, if this model is correct, at least one of these particles will likely be observed at LEPII.

Abstract:
A detailed analysis of dark matter event rates in minimal supergravity models (MSGM) is given. It is shown analytically that the lightest neutralino the ${{\tilde {Z}_{1}}}$ is the LSP over almost all of the parameter space, and hence the natural candidate for cold dark matter (CDM). The radiative breaking of $SU(2)\times U(1)$ constraints are shown to be crucial in determining the expected event rates. Approximate analytic formulae are obtained to determine the gaugino-higgsino content of the ${{\tilde{Z}_{1}}}$ particle.From this one can deduce the behavior of the event rates as one varies the SUSY soft breaking parameters and tan $\beta$. The constraint on the event rates due to the recently measured $b\rightarrow s+\gamma$ decay is calculated. It is seen that this data eliminates most of the parameter space where $\mu$ (the Higgs mixing parameter) and $A_t$ (the t-quark cubic soft breaking parameter) have the same sign. Since the t-quark is close to its Landau pole, $A_t$ is restricted to be mostly positive, and so most of the $\mu>0$ part of the parameter space is eliminated...

Abstract:
We study the predictions of the simplest SU(5) grand unified model within the framework of minimal supergravity, including constraints from the radiative breaking of electroweak symmetry. As a consequence of the unification of the $b$-quark and $\tau$-lepton Yukawa couplings, the top quark mass is predicted to be close to its fixed point value. We delineate the regions of the supergravity parameter space allowed by constraints from the non-observation of proton decay and from the requirement that the LSP does not overclose the universe. These constraints lead to a definite pattern of sparticle masses: the feature unique to Yukawa unified models is that some of the third generation squarks are much lighter than those of the first two generations. Despite the fact that all sparticle masses and mixings are determined by just four SUSY parameters at the GUT scale (in addition to $m_t$), we find that the signals for sparticle production can vary substantially over the allowed parameter space. We identify six representative scenarios and study the signals from sparticle production at the LHC. We find that by studying the signal in various channels, these scenarios may be distinguished from one another, and also from usually studied ``minimal models'' where squarks and sleptons are taken to be degenerate. In particular, our studies allow us to infer that some third generation squarks are lighter than other squarks---a feature that could provide the first direct evidence of supergravity grand unification.

Abstract:
A review of baryon and lepton conservation in supersymmetric grand unified theories is given. Proton stability is discussed in the minimal SU(5) supergravity grand unification and in several non-minimal extensions such as the $SU(3)^3$, SO(10) and some string based models. Effects of dark matter on proton stability are also discussed and it is shown that the combined constraints of dark matter and proton stability constrain the sparticle spectrum. It is also shown that proton lifetime limits put severe constraints on the event rates in dark matter detectors. Future prospects for the observation of baryon and lepton number violation are also discussed.

Abstract:
It is shown that in the physically interesting domain of the parameter space of SU(5) supergravity GUT, the Higgs and the Z poles dominate the LSP annihilation. Here the naive analyses on thermal averaging breaks down and formulae are derived which give a rigorous treatment over the poles. These results are then used to show that there exist significant domains in the parameter space where the constraints of proton stability and cosmology are simultaneously satisfied. New upper limits on light particle masses are obtained.

Abstract:
Prospects for SO(10) as a minimal grand unification group have recently been heightened by several considerations such as the MSW resolution of the solar neutrino puzzle, baryogenesis, possibility for understanding fermion masses etc. I review the present status of the minimal SO(10) models with special emphasis on the predictions for proton lifetime and predictions for neutrino masses for the non-supersymmetric case and discuss some preliminary results for the supersymmetric case. It was generally believed that minimal SO(10) models predict wrong mass relations between the charged fermions of the first and second generations; furthermore, while the smallness of the neutrino masses in these models arises from the see-saw mechanism, it used to be thought that detailed predictions for neutrino masses and mixings require further adhoc assumptions. In this talk, I report some recent work with K.S.Babu, where we discovered that the minimal SO(10) model, both with and without SUSY, has in it a built-in mechanism that not only corrects the bad mass relations between the charged fermions but at the same time allows a complete prediction for the masses and mixings in the neutrino sector. We define our minimal model as the one that consists of the smallest set of Higgs multiplets that are needed for gauge symmetry breaking. Our result is based on the hypothesis that the complex {\bf 10} of Higgs bosons has only a single coupling to the fermions. This hypothesis is guaranteed in supersymmetric models and in non-SUSY models that obey a softly broken Peccei-Quinn symmetry.

Abstract:
A review is given of the historical developments of 1982 that lead to the supergravity unified model (SUGRA)with gravity mediated breaking of supersymmetry. Further developments and applications of the model in the period 1982-85 are also discussed. The supergravity unified model and its minimal version (mSUGRA) are currently among the leading candidates for physics beyond the Standard Model. A brief note on the developments from the present vantage point is included.

Abstract:
GUT scale threshold corrections in minimal SU(5) supergravity grand unification are discussed. It is shown that predictions may be made despite uncertainties associated with the high energy scale. A bound relating the strong coupling constant to the mass scales associated with proton decay and supersymmetry is derived, and a sensitive probe of the underlying theory is outlined. In particular, low energy measurements can in principle determine the presence of Planck scale ($ 1 / {{\rm M}_{\rm Pl}} $) terms.