We have measured infrared spectra from several types of calcite: chalk, freshly cultured coccoliths produced by three species of algae, natural calcite (Iceland Spar), and two types of synthetic calcite. The most intense infrared band, the asymmetric carbonate stretch vibration, is clearly asymmetric for the coccoliths and the synthetic calcite prepared using the carbonation method. It can be very well fitted by two peaks: a narrow Lorenzian at lower frequency and a broader Gaussian at higher frequency. These two samples both have a high specific surface area. Density functional theory for bulk calcite and several calcite surface systems allows for assignment of the infrared bands. The two peaks that make up the asymmetric carbonate stretch band come from the bulk (narrow Lorenzian) and from a combination of two effects (broad Gaussian): the surface or near surface of calcite and line broadening from macroscopic dielectric effects. We detect water adsorbed on the high surface area synthetic calcite, which permits observation of the chemistry of thin liquid films on calcite using transmission infrared spectroscopy. The combination of infrared spectroscopy and density functional theory also allowed us to quantify the amount of polysaccharides associated with the coccoliths. The amount of polysaccharides left in chalk, demonstrated to be present in other work, is below the IR detection limit, which is 0.5% by mass.