By definition, any major petroleum company have major offshore operations--in many cases, quite literally! For example, BP (formerly British Petroleum) owns ARCO and has huge investments in oilfields in Alaska and the Gulf of Mexico--and BP is one of the "smaller" petroleum multinationals compared to bigger ones like Exxon Mobil, Royal Dutch Shell, Chevron, TotalFinaElf, etc.
No, I mean office operations and IT. Accounting, everything. Very little is done now in the USA there.