Experimental, numerical and analytical studies are conducted to gain a better understanding of the influence of the stoichiometric mixture fraction Zst on the structure and critical conditions of extinction of non-premixed methane and non-premixed dimethyl ether flames. Experimental studies were carried out using a counterflow setup, consisting of an oxidizer duct and a fuel duct. A steady, laminar and axisymmetric flow leaves each duct and stagnates against the flow of the opposed duct. In this way a stagnation plane is produced by the two reactant streams, leading to a reaction zone at the boundary layer. By diffusion the oxidant and the fuel form a flammable mixture in the region with the maximum energy output. The combustion in a counter flow burner is mainly dependent on the chemical reaction time and the velocity of the flow of the fuel and the flow of the oxidizer. The characteristic chemical time depends on the adiabatic temperature and the stoichiometric mixture fraction, whereas the characteristic flow time is given by the strain rate. If the fuel and oxidizer velocities exceed a certain value the reaction ends abruptly. This state is called extinction. In order to elucidate the effect of Zst, the mass fractions of the reactants were so chosen that Tst is fixed. For methane, it was found that the strain rate at extinction continually increased with increasing Zst, signifying that with decreasing fuel mass fraction and rising oxygen mass fraction the flame becomes harder to extinguish. This was confirmed by numerical studies and asymptotic analysis. The predictions of the analysis show that with increasing values of Zst, the scalar dissipation rate at extinction Zst,q, first increases and then decreases. A key outcome of the analysis is that with increasing stoichiometric mixture fraction, the thickness of the regions where oxygen and fuel are consumed first increases and then decreases. Numerical computations using the San Diego Mechanism show a full consumption of fuel and a leakage of oxygen for all values of Zst, whereas computations using reduced chemistry show a leakage of fuel from the reaction zone at low values of Zst and a leakage of oxygen at low (1 - Zst). Results of extinction experiments in the counterflow burner with dimethyl ether show a decrease followed by an increase in the strain rate at extinction with increasing values of Zst. This behavior is observed at different oxidizer and fuel mass fractions obtained using various Lewis numbers in the calculation of the mixture fraction. Numerical computations were performed but do not match the experimental results, showing a much more significant decrease in strain rates at extinction.