Is it possible to invest in markets in the USA or Europe from an Asian 3rd-world country over the Internet?
Does the laws of the USA or EU allow people outside to do so?
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Absolutely. It does highly depend on your country, as US brokerages are stricter with or even closed to residents of countries that produce drugs, launder money, finance terror, have traditional difficulty with the US, etc.
It also depends on your country's laws. Some countries have currency controls, restrictions on buying foreign/US securities, etc.
That said, some brokerages have offices world-wide, so there might be one near you. If your legal situation as described above is fortunate, some brokers will simply allow you to setup online using a procedure not too different from US residents: provide identification, sign tons of documents. You'll have to have a method to deliver your documentation in the ways you'd expect: mail, fax, email.
E*Trade is the best starter broker, right now, imo. Just see how far you can go in the sign-up process.