HOW to do it?
The most economical way to circumnavigate is to buy a round-the-world air ticket that uses one airline alliance. Theoretically, any routing is possible, but knowing how the RTW booking system works will make your trip cheaper. For example, the Star Alliance, a coalition of 27 airlines, offers a RTW ticket with a maximum of 15 stops. Its member airlines fly to 1185 airports in 185 countries.
There are rules: you must follow one global direction (east or west — no backtracking); you must start and finish in the same country; and you must book all your flights before departure, though you can change them later (which may incur extra charges).