This chapter explores several of the most common and useful approaches to atmospheric data fitting as well as the process of using air mass factors to produce vertical atmospheric column abundances from line-of-sight slant columns determined by data fitting. An atmospheric spectrum or other type of atmospheric sounding is usually fitted to a parameterized physical model by minimizing a cost function, usually chi-squared. Linear fitting, when the model of the measurements is linear in the model parameters is described, followed by the more common nonlinear fitting case. For nonlinear fitting, the standard Levenberg-Marquardt method is described, followed by the use of optimal estimation, one of several retrieval methods that make use of a priori information to providing regularization for the solution. In the context of optimal estimation, weighting functions, contribution functions, and averaging kernels are described. The Twomey-Tikhonov regularization procedure is presented. Correlated parameters, with the important example of Earth’s atmospheric ozone, are discussed.