This chapter presents the models and processing steps used for Gaia's core solution, namely, the Astrometric Global Iterative Solution (AGIS). The inputs to this solution rely heavily on the basic observables (or astrometric elementaries) which have been pre-processed and discussed in Chapter 2, the results of which have been published in Fabricius et al. (2016). The models consist of reference systems and time scales; assumed linear stellar motion and relativistic light deflection; in addition to fundamental constants and the transformation of coordinate systems. Higher level inputs such as: planetary and solar system ephemeris; Gaia tracking and orbit information; initial quasar catalogues and BAM data are all needed for the processing described here. The astrometric calibration models are outlined followed by the details processing steps which give AGIS its name. The final Section 3.5 represents a basic quality assessment and validation of the scientific results which has also been published in detail in Lindegren et al. (2016). However, the validation of the science products was not restricted to just this, a more independent catalogue consolidation and validation of the science results for Gaia DR1 was also performed and are documented in Chapter 7 and have been published in Arenou et al. (2017).