Methane is a chemically and radiatively important trace gas with a wide range of geographically and temporally varying biogenic and anthropogenic sources and sinks. A powerful method for determining the net methane fluxes due to these sources and sinks involves solution of an inverse problem in which the observed concentrations are effectively lagrangian line integrals and the unknown fluxes are contained in the integrands. The general method calculates optimal estimates in the Bayesian sense using an eulerian or lagrangian atmospheric transport model and global atmospheric methane measurements. We review the results of previous studies of regional and global methane fluxes using these methods. We also present the results of a recently completed study using a Kalman filter and the global 3D Model for Atmospheric Transport and Chemistry (MATCH) driven by NCEP analysed observed meteorology at T62 (1.8o x 1.8o) resolution. Monthly fluxes are optimally estimated for three large wetland/bog regions, rice agriculture (globally aggregated), and three large biomass burning regions. The study uses AGAGE, CMDL and other methane observations for 1996-2001 and also estimates average annual emissions from coal, gas, animals and waste sites. Deduced seasonal cycles for the biogenic sources (including rice) are qualitatively similar to prior estimates from site measurements but show differences in peak amplitudes and phases and significant year-to-year variability. Emissions from rice-growing regions are greater than prior estimates, while emissions from fossil sources are less. Enhanced emissions from northern wetland/bog regions are inferred to be the dominant contributor to the large 1998 increases in methane.