TECHNOLOGY FOCUS - CHEMICAL ENGINEERING - Preferred Apparatus: The reactor used for vapor-phase oxidation is heat-exchanger type fixed bed reactor or fluidized bed reactor using spheroidal catalyst with average particle diameter of 30-100 microns and containing 30-70 wt.% of silica.    TECHNOLOGY FOCUS - INORGANIC CHEMISTRY - Preferred Catalyst: The catalyst is noble metals-supported catalyst of formula: RrB'bN'n.R=palladium, gold or ruthenium;B'=lead, bismuth, thulium or mercury;andN'=magnesium, potassium or sodium.The catalyst is formed by supporting spherical carrier with average particle diameter of 30-200 microns.    TECHNOLOGY FOCUS - ORGANIC CHEMISTRY - Preferred Process: The unreacted unsaturated aldehyde is separate-and-recovered from the reaction liquid containing unsaturated carboxylic acid ester. The unsaturated aldehyde is recycled at the oxidation-esterification process. The unsaturated carboxylic acid byproduced is collected and reacted with alcohol in acidic catalyst to obtain unsaturated carboxylic acid ester. The gas containing alkane and oxygen is contacted with the catalyst at 250degreesC or more to produce unsaturated aldehyde and unsaturated carboxylic acid. The reactive gas obtained is cooled to 200degreesC or less and the component containing the condensate, unsaturated aldehyde, water, unreacted alkane and unsaturated carboxylic acid are separated from the reaction material. Organic solvent containing alcohol, unsaturated aldehyde and unreacted alkane are contacted and water is selectively removed. Non-condensed unsaturated aldehyde is contacted with the organic solvent and the unsaturated aldehyde is collected. The unreacted alkane is collected and recycled to vapor-phase oxidation process. Alcohol solution containing unsaturated aldehyde is contacted with a catalyst in presence of gas containing oxygen to obtain unsaturated carboxylic acid ester. The reaction liquid containing the ester, water, unsaturated carboxylic acid, unreacted unsaturated aldehyde and residual alcohol is separated into distillation composition containing unreacted unsaturated aldehyde which is a low boiling point component and alcohol, and distillation composition containing the ester and acid as high boiling point components and water. The low boiling point component is recycled and the liquid containing high boiling point component is oil/water separated. The ester in the oil layer is purified and ester with high purity is obtained. The acid in the separated liquid is reacted with alcohol in presence of acidic catalyst to obtain ester. SPECIFIC COMPOUNDS - The alkane is propane or isobutane, the unsaturated aldehyde is acrolein or methacrolein, unsaturated carboxylic acid is acrylic acid or methacrylic acid and the ester is acrylic ester or methacrylic ester. The alcohol is methanol, ethanol, n-propanol, isopropanol, n-butanol, sec-butanol, isobutanol, t-butanol, cyclohexanol and/or diethyl hexanol (all claimed).