There isn't one entity that controls the entirety of the internet, thankfully. Although there is one in particular that has a significant amount of power. The Internet Corporation for Assigned Names and Numbers or ICANN is responsible for managing the domain name system (DNS) which pairs domain names with IP addresses. You might be happy to know that the U.S. government isn't in control of ICANN either. ICANN is its own entity.
Controversy
The internet is arguably one of the most powerful things in the world. So of course international governments have their own ideas of how it should be controlled. There are countries that want to get rid of ICANN and give each country complete control over all internet activities.
On the other hand, there are countries that want to use a multinational stakeholder system which would keep everyone accountable to certain standards of use. Then there are other arguments on whether the U.N. should step in and play a role or not.
You can be sure of one thing though. When the U.S. gave up its oversight of ICANN, the international community sighed a bit of relief. At least there's not as big of a threat of one government being able to control the internet or do surveillance on other countries without them knowing.
The other entities
A few other entities have considerable authority over the standards that we use on the internet. There's the Internet Engineering Task Force (IETF) and it works to create new standards for the internet. The standards they set are all voluntary for people to adhere to, but in general most people do.
Internet standards include all protocols, procedures, and conventions that are used on the internet. For example, TCP/IP is one of the standards the IETF is responsible for. The IETF is really like a review committee. They get new standards made by different corporations and people and they review them to make sure they are going to work well across international systems.
Next is the W3C, but first we need to clear up the difference between the world wide web and the internet (no, they aren't the same). The internet is a network of computers that communicate with each other using TCP/IP. The world wide web is where all of the websites are and it uses URIs (uniform resource identifiers) to locate those websites.
When you think of the world wide web, think about URLs, HTTP, and HTML. That's what the W3C works to maintain. They set the standards for web design, HTML semantics, XML, SOAP, and how accessible websites are for people with disabilities and on mobile devices.
One more entity is the Internet Assigned Numbers Authority (IANA). They are a part of ICANN and they work closely with IEFT. The IANA is responsible for coordinating domain names, protocol assignments, and number resources. So they manage the DNS Root, the global pool of IP and AS numbers, and the different codes and numbers for internet protocols.
IANA works with IEFT to handle the internet protocols and it works with ICANN to handle the DNS stuff. As far as the internet goes, IANA is one of the oldest entities, started in the 1970s.
This is the basic rundown of who governs the internet. Thankfully it isn't one entity and it isn't any specific government. The different entities work together to make sure the internet stays as open and free as it possibly can. Hopefully we can keep it like that.
As a side note, have you ever gone to the W3 site to see any of the new HTML standards? This is where you'll see all of the details about how HTML has evolved over the years and it's kind of interesting.
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