Hydrogen atom abstraction from allylic C-H bonds by molecular oxygen plays a very important role in determining the reactivity of fuel molecules having allylic hydrogen atoms. Rate constants for hydrogen atom abstraction by molecular oxygen from molecules with allylic sites have been calculated. A series of molecules with primary, secondary, tertiary, and super secondary allylic hydrogen atoms of alkene, furan, and alkylbenzene families are taken into consideration. Those molecules include propene, 2-butene, isobutene, 2-methylfuran, and toluene containing the primary allylic hydrogen atom; 1-butene, 1-pentene, 2-ethylfuran, ethylbenzene, and n-propylbenzene containing the secondary allylic hydrogen atom; 3-methyl-1-butene, 2-isopropylfuran, and isopropylbenzene containing tertiary allylic hydrogen atom; and 1-4-pentadiene containing super allylic secondary hydrogen atoms. The M06-2X/6-311++G(d,p) level of theory was used to optimize the geometries of all of the reactants, transition states, products and also the hinder rotation treatments for lower frequency modes. The G4 level of theory was used to calculate the electronic single point energies for those species to determine the 0 K barriers to reaction. Conventional transition state theory with Eckart tunnelling corrections was used to calculate the rate constants. The comparison between our calculated rate constants with the available experimental results from the literature shows good agreement for the reactions of propene and isobutene with molecular oxygen. The rate constant for toluene with O2 is about an order magnitude slower than that experimentally derived from a comprehensive model proposed by Oehlschlaeger and coauthors. The results clearly indicate the need for a more detailed investigation of the combustion kinetics of toluene oxidation and its key pyrolysis and oxidation intermediates. Despite this, our computed barriers and rate constants retain an important internal consistency. Rate constants calculated in this work have also been used in predicting the reactivity of the target fuels of 1-butene, 2-butene, isobutene, 2-methylfuran, 2,5-dimethylfuran, and toluene, and the results show that the ignition delay times for those fuels have been increased by a factor of 1.5-3. This work provides a first systematic study of one of the key initiation reaction for compounds containing allylic hydrogen atoms.