Forest wildfires pose significant and growing threats to human safety, wildlife habitat, regional economies and global climate change. It is crucial that forest fires be subject to timely and accurate monitoring by forest fire managers and other stake-holders. Measurement by spaceborne equipment has become a practical and appealing method to monitor the occurrence and development of forest wildfires. Here we present an overview of the principles and case studies of forest fire monitoring (FFM) with satellite- and drone-mounted infrared remote sensing (IRRS). This review includes four types of FFM-relevant IRRS algorithms: bi-spectral methods, fixed threshold methods, spatial contextual methods, and multi-temporal methods. The spatial contextual methods are presented in detail since they can be applied easily with commonly available satellite IRRS data, including MODIS, VIIRS, and Landsat 8 OLI. This review also evaluates typical cases of FFM using NOAA-AVHRR, EOS-MODIS, S-NPP VIIRS, Landsat 8 OLI, MSG-SEVIRI, and drone infrared data. To better implement IRRS applications in FFM, it is important to develop accurate forest masks, carry out systematic comparative studies of various forest fire detection systems (known as forest fire products), and improve methods for assessing the accuracy of forest fire detection. Medium-resolution IRRS data are effective for landscape-scale FFM, and the VIIRS 375 m contextual algorithm and RST-FIRES algorithm are helpful for closely tracking forest fires (including small and short-lived fires) and forest-fire early warning.