The effects of compression wave and flame stretch on the determination of flame speeds using spherical bombs under constant pressure and constant volume conditions are studied theoretically and numerically. A time accurate and front adaptive numerical algorithm is developed to simulate the outwardly propagating spherical flame in a closed chamber in a broad range of pressures and equivalence ratios. The results show that both compression wave and flame stretch have significant impacts on the accuracy of measured flame speeds. For the constant pressure method, a new correlation to obtain compression corrected flame speed (CCFS) in nearly constant pressure spherical bomb experiments is presented. For the constant volume spherical bomb experiments, a method to obtain stretch corrected flame speed (SCFS) at elevated pressures and temperatures is also presented. The results show that the present methods can dramatically increase the reliability of the measured flame speeds and the pressure range of flame speed measurements in experiments.