O O || || C - CH3 + ROH <---> C - CH3 + CH3OH / / CH3O RO
(ester + alcohol <---> different ester + different alcohol)Acids can catalyse the reaction by donating a proton to the alkoxy group, thus making it more reactive, while bases can catalyse the reaction by removing a proton from the alcohol, thus making it more reactive.
History of biolipid transesterification
One of the first uses of transesterified vegetable oil (biodiesel) was powering heavy duty vehicles in South Africa before World War II. The name "biodiesel" has been given to transesterified vegetable oil to describe its use as a diesel fuel.
It was patented in the US in the 1940s by Colgate (and other) scientists, though biolipid transesterification might have been discovered much earlier. The 1940s researchers were looking for a method to produce glycerine more readily, in order to produce explosives for World War II. Many of the methods used today by producers and homebrewers have their origin in the original 1940s research.