Hydrogen as an energy carrier is a very promising alternative fuel in the future. Accidental hydrogen explosions remain one of the major concerns in hydrogen energy utilization and process industries. This paper summarizes recent experimental and numerical efforts towards understanding combustion wave propagation in hydrogen explosions, including flame instabilities, flame acceleration, deflagrations, and deflagration-to-detonation transition (DDT). The fundamental problems involve understanding physical mechanisms that significantly influence the dynamic flame behavior in hydrogen explosions, such as combustion/hydrodynamic instabilities, vortex motion, pressure waves and flow turbulence. Advances achieved over recent years in new experimental observations, theoretical models and numerical simulations are discussed. Future research is required to quantitatively understand flame instabilities, turbulence properties and DDT in hydrogen explosions and improve reliability of theoretical and numerical predictions for hydrogen safety applications.