Accelerated curing of high performance fibre-reinforced polymer (FRP) composites via microwave heating or radiation, which can significantly reduce cure time and increase energy efficiency, has several major challenges (e.g. uneven depth of radiation penetration, reinforcing fibre shielding, uneven curing, introduction of hot spots etc). This article reviews the current scientific challenges with microwave curing of FRP composites considering the underlying physics of microwave radiation absorption in thermoset-matrix composites. The fundamental principles behind efficient accelerated curing of composites using microwave radiation heating are reviewed and presented, especially focusing on the relation between penetration depth, microwave frequency, dielectric properties and cure degree. Based on this review, major factors influencing microwave curing of thermoset-matrix composites are identified, and recommendations for efficient cure cycle design are provided.