The SM, conceptually and phenomenologically fails to incorporate and explain few fundamental problems of particle physics and cosmology, such as a viable dark matter candidate, mechanism for inflation, neutrino masses, the hierarchy problem etc. In addition, the recent discovery of the 125 GeV Higgs boson and the top quark mass favor the metastablility of the electroweak vacuum, implying the Higgs boson is trapped in a false vacuum. In this thesis we propose the simplest extension of the SM by adding an extra degree of freedom, a scalar singlet. The singlet can mix with the Higgs field via the Higgs portal, and as a result we obtain two scalar mass eigenstates (Higgs-like and singlet-like). We identify the lighter mass eigenstate with the 125 GeV SM Higgs boson. Due to the mixing, the SM Higgs quartic coupling receives a finite tree level correction which can make the electroweak vacuum completely stable. We then study the stability bounds on the tree level parameters and determine the allowed mass region of the heavier mass eigenstate (or singlet-like) for range of mixing angles where all the bounds are satisfied. We also obtain regions of parameter space for different signs of the Higgs portal coupling. In the allowed region, the singlet-like state can decay into two Higgs-like states. We find the corresponding decay rate to be substantial. Finally, we review various applications of the singlet extension, most notably, to the problem of dark matter and inflation.
