Selective oxidation or oxidative functionalization of methane and ethane by both homogeneous and heterogeneous catalysis is presented concerning: (1) selective oxidation of methane and ethane to organic oxygenates by hydrogen peroxide in a water medium in the presence of homogeneous osmium catalysts, (2) selective oxidation of methane to formaldehyde over highly dispersed iron and copper heterogeneous catalysts, (3) selective oxidation of ethane to acetaldehyde and formaldehyde over supported molybdenum catalysts, and (4) oxidative carbonylation of methane to methyl acetate over heterogeneous catalysts containing dual sites of rhodium and iron. ; National Natural Science Foundation of China [20433030, 20625310, 20773099, 20873110]; National Basic Program of China [2005CB221408, 2010CB732303]; Key Scientific Project of Fujian [2009HZ0002-1]; Program for New Century Excellent Talents in Fujian