A process for continuously producing acetic anhydride alone or acetic anhydride and acetic acid by reacting methyl acetate and/or dimethyl ether and, optionally, water and/or methanol, with carbon monoxide alone or carbon monoxide and hydrogen in the presence of a rhodium compound and methyl iodide as principal catalysts. Trace impurities causative of tar formation are distilled and separated in an evaporator and/or a subsequent refining step to remove the same. The removal of the trace impurities causative of tar formation serves to decrease the amount of tar formed as an impurity.