It was noticed in [482] that the potential for N = 2 SUSY gauge theory at ξ$_{+}$ = $^{ξ}$− = 0, ξ$_{3}$ = | ξ | are similar to the case of D -term inflation studied before in [645] with W = λS Φ$_{+}$$^{Φ}$− and D = | Φ$_{+}$ | 2 − | $^{Φ}$− | 2 − ξ , for which the potential is given by: with an assumption, λ = √$_{2}$$_{g}$$_{. If instead,}$ ξ$_{+}$ = $^{ξ}$− = 2 M 2 = ξ , one recovers a potential of an F -term inflation model with, W = λS (Φ ′$_{+}$ Φ ′ − − M $^{2}$), and, D = | Φ $^{′$_{+}$}$| 2 − | Φ ′ − | $^{2}$, with a potential: The first term of the potential now coincides with the F -term inflation potential in N = 1 theory, proposed by [62], under the assumption that the gauge coupling g = $_{λ/}$√$_{2.}$ The P -term model could be coupled to N = 1 SUGRA and the inflationary potential is given by [462, 482] The inflating trajectory takes place at Φ$_{+}$ $^{= Φ}$− = 0. After adding the 1-loop correction from gauge fields, one obtains [482] where f = ξ $_{1}$+ ξ 2 ξ 2 , 0 ≤ f ≤ 1 and . . . stands for terms | S | $^{6}$/ 2 M 6 p and higher order gravitational corrections. Special case f = 0 corresponds to D-term inflation, and f = 1 corresponds to F-term inflation. A general P-term inflation model has an arbitrary 0 ≤ f ≤ 1. The running of the spectral index from blue to red and the amplitude of the perturbations are similar to what has been studied in [513]. Embedding inflation within GUT has a long history. We give here an overview of some of the old attempts and the current status of some of these models, and then turn to more recent developments. G. Embedding inflation in SUSY GUTs 128 V D N =2 = 2 g 2 ( | S Φ$_{+}$ | 2 + | S $^{Φ}$− | 2 + | Φ$_{+}$$^{Φ}$− | 2 ) + g 2 2 ( | Φ$_{+}$ | 2 − | $^{Φ}$− | 2 − ξ ) 2 . (278) V F N =2 = 2 g 2 ( | S $^{$_{Φ}$′$_{+}$}$| 2 + | S $_{Φ}$′ − | 2 + | $_{Φ}$′$_{+}$$_{Φ}$ ′ − − M $^{2}$|2 ) + g 2 2 ( | $^{$_{Φ}$′$_{+}$}$| 2 − | $_{Φ}$′ − | 2 ) 2 . (279) V = 2 g $^{2}$e$^{|}$$^{S}$$^{|}$$^{2}$ M 2 p [ | Φ$_{+}$$^{Φ}$− − ξ$_{+}$/ 2 | 2 ( 1 − S ¯ S M 2 p + ( S ¯ S M 2 p ) 2 ) + | S Φ$_{+}$ | 2 + | S $^{Φ}$− | 2 ] + g 2 2 ( | Φ$_{+}$ | 2 − | $^{Φ}$− | 2 − ξ$_{3}$ ) 2 . (280) V = g $^{2}$ξ2 2 ( 1 + g 2 8 π 2 ln | S $^{2}$| S 2 c | + f | S | 4 2 M 4 p + . . . ) , (281)