The Faint Infrared Grism Survey (FIGS) is a deep Hubble Space Telescope (HST) WFC3/IR (Wide Field Camera 3 Infrared) slitless spectroscopic survey of four deep fields. Two fields are located in the Great Observatories Origins Deep Survey-North (GOODS-N) area and two fields are located in the Great Observatories Origins Deep Survey-South (GOODS-S) area. One of the southern fields selected is the Hubble Ultra Deep Field. Each of these four fields were observed using the WFC3/G102 grism (0.8 mu m-1.15 mu m continuous coverage) with a total exposure time of 40 orbits (approximate to 100 kilo-seconds) per field. This reaches a 3 sigma continuum depth of approximate to 26 AB magnitudes and probes emission lines to similar to 10(-17) erg s(-1) cm(-2). This paper details the four FIGS fields and the overall observational strategy of the project. A detailed description of the Simulation Based Extraction (SBE) method used to extract and combine over 10,000 spectra of over 2000 distinct sources brighter than m(F105W) = 26.5 mag is provided. High fidelity simulations of the observations is shown to significantly improve the background subtraction process, the spectral contamination estimates, and the final flux calibration. This allows for the combination of multiple spectra to produce a final high quality, deep, 1D spectra for each object in the survey.