F -terms, describing interaction between chiral superfields, and the second from the D -terms, which contains the gauge interactions. The scalar potential derived from W and K has to be non-vanishing to support inflation, therefore breaking (transiently) SUSY. Therefore two classes of models driven by F -terms or by D -terms have been proposed. We will discuss first old and new properties of F -and D -term hybrid inflation, which is by far the most popular amongst the model builders. Some of these models have also been reviewed in Refs. [10, 611]. 1. Chaotic inflation in SUSY Embedding chaotic inflation within SUSY is a challenging problem, discussed long ago and is still under investigation by many [494, 505–507, 612–614]. As discussed in Sec. IV B 1, chaotic inflation in its non-SUSY description requires super-Planckian VEV to last long enough. By construction SUSY and SUGRA are only valid when φ ≪ M$_{P}$ . Another challenge comes from the fact that chaotic inflation requires a UV-complete theory that SUGRA does not provide [494]. Even discarding the question of validity, if inflation was driven by the F -terms, the expo- nential of the K¨ ahler potential potential can generically spoil the flatness of the potential. To tackle this problem, non-minimal K¨ ahler potentials with a logarithmic form have been proposed [505], which can be combined to a mass-term superpotential, W = M Φ $^{2}$, to gen- erate the right scalar potential, driven by F -terms. It has also been shown that a chaotic potential can also be embedded within D -terms [506, 507]. They assume 4 chiral superfields and that the symmetries of the theory are U (1)$_{gauge}$ × U (1)$_{R}$ $^{57}$. A renormalizable super- potential and a minimal K¨ ahler potential can generate a scalar potential possessing many F - flat valleys lifted by D -terms required for generating chaotic inflation. It was pointed out in Ref. [614] that the mechanism requires a fine-tuned moduli sector which is hard to embed within string theory. As a conclusion, the question of managing super-Planckian VEVs and densities within SUGRA does not make chaotic models appealing from particle 57 A challenging problem for these kind of attempts is that the inflaton sector, though it resides in a hidden sector, must couple to the observable sector, i.e. MSSM. This modifies the D -flatness conditions for any extra U (1)$_{gauge}$ sector added to the SM gauge group, which opens up new D -flat directions and this will eventually modify the inflationary potential by lifting such combinations with the help of F -term. Such contributions are often ignored in the literature [506, 507, 614]. 99