This was discussed at random points in a few of my classes, but there are all sorts of issues that are coming regarding the way computer IP addresses are put together. It became clear in the 1990's that we were eventually going to run out of these addresses. To fix it, a new standard called IPv6 was introduced (the one in use is called IPv4). This "new" standard was introduced in 1996.
If there is a solution, you might ask, why has there been no movement towards the new standard? The answer is inertia. In other words "My system works, that will be someone else's problem".
So, that problem looks to be coming up soon, and unless someone comes up with a good solution soon, it will lead to either a slower Internet connection (servers will have to work harder) or a bunch of people being forced to upgrade their networking equipment (unlikely).
I ran across this article, and it's a few months old but still relevant. This article estimates that the day where we "run out of" IP addresses could come as soon as 2010.