Autoignition and oxidation of two Methane (CH4) and Dimethyl Ether (CH3OCH3 or DME) mixtures in air were studied in shock tubes over a wide range of equivalence ratios at elevated temperatures and pressures. These experiments were conducted in the reflected shock region with pressures ranging from 0.8 to 35.7 atmospheres, temperatures ranging from 913 to 1650 K, and equivalence ratios of 2.0, 1.0, 0.5, and 0.3. Ignition delay times were obtained from shock-tube endwall pressure traces for fuel mixtures of CH4/CH3OCH3 in ratios of 80/20 percent volume and 60/40 percent volume, respectively. Close examination of the data revealed that energy release from the mixture is occurring in the time between the arrival of the incident shock wave and the ignition event. An adjustment scheme for temperature and pressure was devised to account for this energy release and its effect on the ignition of the mixture. Two separate ignition delay correlations were developed for these pressure- and temperature-adjusted data. These correlations estimate ignition delay from known temperature, pressure, and species mole fractions of methane, dimethyl ether, and air (0.21 O2 + 0.79 N2). The first correlation was developed for ignition delay occurring at temperatures greater than or equal to 1175 K and pressures ranging from 0.8 to 35.3 atm. The second correlation was developed for ignition delay occurring at temperatures less than or equal to 1175 K and pressures ranging from 18.5 to 40.0 atm. Overall good agreement was found to exist between the two correlations and the data of these experiments. Findings of these experiments also include that with pressures at or below ten atm, increased concentrations of dimethyl ether will consistently produce faster ignition times. At pressures greater than ten atmospheres it is possible for fuel rich mixtures with lower concentrations of dimethyl ether to give the fastest ignition times. This work represents the most thorough shock tube investigation for oxidation of methane with high concentration levels of dimethyl ether at gas turbine engine relevant temperatures and pressures. The findings of this study should serve as a validation for detailed chemical kinetics mechanisms.