It is now a bit more than 18 years since Frank Kastenholtz first noticed that assignments of one class of IP addresses would soon outrun supply. This observation led the press to think that the Internet was running out of all IP addresses (it was not) and to the IETF initiating a process to develop a "next generation" Internet protocol that would have vastly more addresses so that it would, in theory, never run out.
A lot of things have changed in the Internet since 1992 but, finally, we are approaching the time when there will be no new IP addresses to assign to new Internet users. At least there will be no more of the kind of addresses the Internet has been using for most of its existence.
Kastenholtz's observation led the IETF to develop IP version 6 (IPv6). It also led the IETF to recommend changing the way IP addresses were assigned to be more conservative of address space. Kastenholtz, along with other researchers, predicted in 1994 that we would run out of IP version 4 addresses in 2008 plus or minus three years. That prediction turns out to be quite good, the current prediction is that the repository of IPv4 addresses will run out next June (right at 2008+3) and the regional IP address registries will run out less than a year later.
The lack of IPv4 addresses to assign does not mean that the Internet will stop working or growing. Organisations that have unused IPv4 addresses will be able to transfer them to organisations that need addresses. But IPv4 addresses will become harder, and more expensive, to get. The better long term solution is to migrate to IPv6.
Major operating systems and equipment vendors already support IPv6 but there are not all that many IPv6-enabled Internet sites or service providers in the United States. That is slowly changing. Google, Facebook and other big service providers are bringing up IPv6 sites and some of the large ISPs can transport IPv6. My own site is not yet on IPv6 although my Mac-based systems have supported IPv6 for years. My ISP (Comcast) only recently started IPv6 trials. I expect to be IPv6-connected as soon as Comcast rolls out the service.
If you work for a company that has an Internet presence you should be thinking about bringing up your websites on IPv6, as well as IPv4, so that future IPv6-only customers will be able to access your services.
We are running out of IPv4 addresses because the Internet is a success. There are now more than 1.9 billion Internet users in the world. They use more than 760 million computers and they access over 200 million websites. It is no wonder that few of the 4 billion IPv4 addresses are left for assignment. IPv6 increases the number of addresses by a factor of 4 billion so this transition process will not be needed again for a very long time.
The Internet will support IPv4 addresses for a very long time, so if you do not IPv6-enable your environment it will keep working. But over time a smaller percentage of the Internet will be able to reach your services.
Disclaimer: As a board member of ARIN (the regional Internet registry for America, Canada and some of the Caribbean), I watch the run-out process with much interest and, as co-director of the IETF's IPv6 project, I watch the IPv6 adoption process with at least as much interest. But Harvard has not stated any University opinions on the topics, so the above discussion is my own.