These cover the bulk of countries outside Europe. Forex brokers catering for India, Hong Kong, Qatar etc are likely to have regulation in one of the above, rather than every country they support. Some brands are regulated across the globe (one is even regulated in 5 continents). Some bodies issue licenses, and others have a register of legal firms.
Non-bank foreign exchange companies offer currency exchange and international payments to private individuals and companies. These are also known as "foreign exchange brokers" but are distinct in that they do not offer speculative trading but rather currency exchange with payments (i.e., there is usually a physical delivery of currency to a bank account).
However, since the Forex market is a global market, it means there is always a part of the world that is awake and conducting business, and during these hours their currencies tend to experience the most movement. For example, currency pairs involving the US dollar experience the most movement during US business hours (16:00 to 24:00 GMT), while the Euro, Pound, Swiss Franc and other European currencies experience the most movement during European business hours, (8:00 and 16:00 GMT).
Unlike stock markets, which can trace their roots back centuries, the forex market as we understand it today is a truly new market. Of course, in its most basic sense - that of people converting one currency to another for financial advantage - forex has been around since nations began minting currencies. But the modern forex markets are a modern invention. After the accord at Bretton Woods in 1971, more major currencies were allowed to float freely against one another. The values of individual currencies vary, which has given rise to the need for foreign exchange services and trading.