A substrate (e.g., a cycloalkane, a polycyclic hydrocarbon, an aromatic compound having a methyl group) is oxidized with oxygen in the presence of an oxidation catalytic system comprising an imide compound of the following formula (1) (e.g., N-hydroxyphthalimide) and a co-catalyst containing a element selected from Group 3 to 12 elements (in particular, Group 4 to 11 elements) of the Periodic Table of Elements. The co-catalyst comprises a compound containing plural elements (except heteropolyacid and a combination of Group 7 and 8 elements of the Periodic Table of Elements), and is useful for the formation of an oxide (e.g., a ketone, an alcohol, a carboxylic acid): ##STR1## wherein R.sup.1 and R.sup.2 represent a substituent such as a hydrogen atom or a halogen atom, or R.sup.1 and R.sup.2 may together form a double bond or an aromatic or nonaromatic 5- to 12-membered ring, X is O or OH, and n is 1 to 3.