The machine described in this document is an advanced Source of up to 20 MeV Gamma Rays based on Compton back-scattering, i.e. collision of an intense high power laser beam and a high brightness electron beam with maximum kinetic energy of about 720 MeV. Fully equipped with collimation and characterization systems, in order to generate, form and fully measure the physical characteristics of the produced Gamma Ray beam. The quality, i.e. phase space density, of the two colliding beams will be such that the emitted Gamma ray beam is characterized by energy tunability, spectral density, bandwidth, polarization, divergence and brilliance compatible with the requested performances of the ELI-NP user facility, to be built in Romania as the Nuclear Physics oriented Pillar of the European Extreme Light Infrastructure. This document illustrates the Technical Design finally produced by the EuroGammaS Collaboration, after a thorough investigation of the machine expected performances within the constraints imposed by the ELI-NP tender for the Gamma Beam System (ELI-NP-GBS), in terms of available budget, deadlines for machine completion and performance achievement, compatibility with lay-out and characteristics of the planned civil engineering.