The organisation said it had already been contacted about setting up domains in the Cyrillic script - used in many Eastern European countries.
"This is a huge step forward in the development of the internet - it will unblock something that has prevented a lot of people getting online," said Emily Taylor, director of legal and policy at Nominet, the national registry for .uk domain names.
"At the moment, there are one-and-a-half billion people online and four-and-a-half billion people for whom the Roman script just means nothing."
Dr Paul Twomey, chief executive of Icann, described passing the resolution as a "historic moment".
Icann has been working towards opening up the 25-year-old net addresses for nearly six years. It was one of its founding goals in 1998.
At the moment, top-level domains (TLDs) are currently limited to individual countries, such as .uk (UK) or .it (Italy), as well as to commerce, .com, and to institutional organisations, such as .net, or .org.
The .com suffix is the most popular and most costly TLD.
To get around the restrictions, some companies have used the current system to their own ends.
For example, the Polynesian island nation Tuvalu has leased the use of the .tv address to many television firms.
Under the new plans, domain names can be based on any string of letters, in any script.
Individuals will be able to register a domain based on their own name, for example, as long as they can show a "business plan and technical capacity".
Companies will be able to secure domain names based on their intellectual property.
The result could be thousands or even millions of new addresses.
"The most likely new TLDs to be pushed into the Icann process are those that have been under development for some time now - the geo-TLDs such as .cym for Wales, .sco for Scotland, .ldn for London, .nyc for New York and so on," explained Marcus Eggensperger, of Lycos Webhosting.
However, the cost of setting up a domain - at least initially - will be an expensive business.
"We expect that the fee will be in the low six figure dollar amounts," said Dr Twomey.
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