The changing composition of the earth’s atmosphere is a matter of intense scientific research as we strive to understand details of the physical and chemical mechanisms that control our climate. Fourier transform spectroscopy has been applied very successfully to the study of trace gases in the atmosphere by examining terrestrial atmospheric absorption lines in the infrared spectrum from the Sun. In fact many gases were first discovered in the atmosphere during the 1940’s from their absorption features in the infrared solar spectrum. These early optical absorption measurements of the atmosphere using the Sun as a source were made with grating spectrometers and examples of atmospheric gases first detected this way include methane and CO [Migeotte, 1948; 1949]. Continuous or semi-continuous records of infrared solar atmospheric absorption spectra have been made from ground-based Fourier transform spectrometers (FTS) since the late 1970s and early 1980s, when the first ground-based solar-tracking FTS systems were installed at Kitt Peak National observatory in the USA and at the Jungfraujoch Observatory in Switzerland. Initially interest was focused on the detection and quantification of stratospheric trace gases [Rinsland et al., 1986; Zander et al., 1986]. The discovery of the Antarctic ozone hole [Farman et al., 1985] intensified interest in stratospheric chemistry and helped support the establishment of the Network for Detection of Stratospheric Change (NDSC). This global network of instrument sites became operational in 1991 with ground-based FTS amongst the suite of primary techniques being used. Photographs of the instrument at the NDACC site at Wollongong, Australia are shown for illustrative purposes in figure 1 below. Other NDSC instruments are lidars for ozone, temperature, water and aerosols; microwave instruments for ozone, water and chlorine monoxide; UV/Visible spectrograph for ozone and nitrogen dioxide; Dobson/Brewer spectrophotometers for total column ozone and regular ozone sondes. This resulted in a huge increase in the number of infrared solar absorption measurements being made around the globe during the next few years, e.g. [Bell et al., 1994; Bell et al., 1996; Bell et al., 1998; Blumenstock et al., 1997; David et al., 1993; Griffith et al., 1998; Jones et al., 1994; Liu et al., 1992; Mahieu et al., 1995; Notholt, 1994; Notholt et al., 1997; Toon et al., 1999; Toon et al., 1995; Zander et al., 1994]. More recently interest in atmospheric chemistry has been focused on tropospheric pollution and anthropogenic emissions of greenhouse gases [Barret et al., 2003; Jones et al., 2009; Mahieu et al., 1995; Nagahama et al., 2007; Paton-Walsh et al., 2008; Rinsland et al., 2000; Rinsland et al., 2001; Rinsland et al., 2002; Rinsland et al., 2008; Warneke et al., 2006; Zhao et al., 2000; Zhao et al., 2002]. As a result, the NDSC has changed its emphasis and name to the Network for Detection of Atmospheric Composition and Change (NDACC) – see http://www.ndacc.org/. As well as an ever increasing number of sites in the global network the new millennium has seen an expansion into the near infrared spectra region in an effort to provide extremely accurate and precise measurements of carbon dioxide. The Total Column Carbon Observing Network (TCCON) was established to help characterise biogenic and oceanic sources and sinks of greenhouse gases to and from the atmosphere and to validate current and future satellite based measurements (http://www.tccon.caltech.edu/ ). In this chapter the reader will get a brief introduction to the basic theory behind the retrieval of atmospheric trace gas amounts from atmospheric solar infrared transmission spectra and an overview of the previous successes and current challenges in this field of research.