It seems some web architects aim to have a stateless web application. Does that mean basically not storing user sessions? Or is there more to it?
If it is just the use
NOTE: session data is really cached data. This is what it should be used for. If you have an expensive query which is going to be reused, then save it into session. Just remember that you cannot assume it will be there when you try and get it later. Always check if it exists before retrieving.
From a developer's perspective, statelessness can help make an application more maintainable and easier to work with. If I know a website I'm working on is stateless, I need not worry about things being correctly initialized in the session before loading a particular page.
From a user's perspective, statelessness allows resources to be linkable. If a page is stateless, then when I link a friend to that page, I know that they'll see what I'm seeing.
From the scaling and performance perspective, see tsters answer.