A wide range of scientific questions regarding chemically and/or radiatively important trace gases necessitate determinations of their sources and sinks at local to global scales. A powerful method for such determinations involves solution of an inverse problem in which the observed concentrations are effectively Lagrangian line integrals and the unknown sources or sinks are contained in the integrands. The inverse problem consists of calculating optimal estimates of the unknowns in the Bayesian sense using an atmospheric transport model and trace gas measurements gathered over space and time. Great care is necessary to include the effects of both measurement and transport model errors in calculating the uncertainty in the optimal estimates. We review the results of recent studies which use three-dimensional Eulerian (specifically MATCH) or Lagrangian transport models and Kalman filter and other optimization methods to compute emissions of methane, nitrous oxide, and selected halocarbons. These studies use high frequency trace gas observations from global networks (AGAGE, CMDL) to calibrate a priori emission maps for particular processes and geographic regions. The methods allow estimation of time varying emissions. For the hydrogen-containing gases these emission estimates require accurate specification of the concentrations of the hydroxyl radical which constitute their major sink. Hydroxyl radical levels can be optimally estimated in a separate problem using measurements of methyl chloroform whose global emissions are already very well known. The results show that the inverse approach is a powerful complement to traditional surface flux aggregation methods. At the same time, the inverse approach has its own limitations associated especially with transport model errors and/or inadequate atmospheric measurements.