It is evident that the origin of EGRB is not well established. In this thesis I examine the unresolved discrete origin of EGRB. The contribution from normal galaxies, starburst galaxies and AGNs to the EGRB is examined. The second chapter includes the methodology used to find the contributions from different source classes. In the third chapter the contribution from normal and starburst galaxies is discussed. A methodology is developed to derive the contribution from normal and starburst galaxies to the EGRB considering all the major γ-ray production processes in these galaxies. The calculations in this thesis consider the detailed γ-ray measurements of our galaxy(Hunter etal.1997) to derive suitable scaling relations to extend the analysis beyond the Milky Way. It is assumed that all normal and starburst galaxies also have similar γ-ray spectra. A relationship is derived between the γ-ray luminosity and SFR of a normal galaxy. Infrared luminosity of a normal galaxy is used as a tracer of SFR of that galaxy (Kewley et al.2002). For starburst galaxies, the contribution depends on the relative ratio(β)of cosmic-ray enhancement per SFR w.r.t the Milky Way. To find the proportionality constants between cosmic-ray production rate and SFR of starburst galaxies, M82 has been taken as a standard. Contribution from FSRQs and BL Lacs to the EGRB is discussed in the fourth chapter. FSRQs and BL Lacs are considered as separate source classes, and their luminosity functions are constructed separately from the recent identifications of EGRET sources(Sowards-Emmerd,Romani&Michelson2003 and Sowards-Emmerd et al.2004) which almost doubled the blazers count than that used by Chiang & Mukher-jee(1998). Radio-loud AGNs with all possible jet to line-of-sight angle (SSRQs, FR IIs, FR Is) are termed here off-axis AGNs. It is considered that SSRQs and FSRQs and FR IIs are from one parent population, BL Lacs and FR Is are from another parent population. The scenario considered includes an AGN jet which slows down as it moves away from the central source. The contributions from these AGNs (relative to the FSRQs and BL Lacs contribution) are discussed in chapter five. Chapter six briefly summarised the findings from the thesis.