High-resolution imaging spectroscopy has enabled the detection and quantification of methane emissions from individual facilities, providing valuable information for identifying large emission sources and evaluating emission inventories. This review synthesizes research on broad-spectrum imaging spectrometers through an end-to-end framework linking spectral characterization, methane enhancement retrieval, plume segmentation and emission-rate estimation. We first review atmospheric radiative transfer and spectral characterization, followed by methane enhancement retrieval methods, including IMAP-DOAS, matched filters, and their variants. We then discuss the identification of methane plumes, with particular attention to false-positive suppression and recent developments in automated plume segmentation. Emission-rate estimation methods, including Gaussian plume inversion, cross-sectional flux, integrated mass enhancement, and machine learning approaches, are compared in terms of their underlying assumptions and sources of uncertainty. We also examine the probability of detection and emission persistence, which are essential for interpreting nondetections and estimating emissions from repeated observations. Applications across oil and gas infrastructure, coal mines, landfills, offshore facilities, and accidental releases demonstrate the capability of imaging spectroscopy to identify major emission sources and reveal their spatial and temporal variability. Continued advances in spectral calibration, methane enhancement retrieval, automated plume identification, and emission-rate estimation, together with coordinated multisensor observations, will further improve the accuracy, consistency, and temporal coverage of facility-scale methane monitoring.