Qt (pronounced as /ˈkjuːt/ "cute") is a cross-platform application framework that is widely used for developing application software with a graphical user interface (GUI) (in which cases Qt is classified as a widget toolkit), and also used for developing non-GUI programs such as command-line tools and consoles for servers. 
In Gentoo, the Qt ebuilds and related packages are maintained by a dedicated team which is a subproject of the Gentoo Desktop project. This team also maintains the official qt overlay. They are very open to user contributions and volunteers who want to become new developers.
Installing the Qt libraries
Most users will not need to install the Qt libraries intentionally, but can just let them be pulled in as dependencies of the applications they want to use. For those who want the complete set (e.g. for software development purposes), there is an x11-libs/qt-meta ebuild, and there are sets in the overlay.
Qt based desktop environments
There are currently two Qt based desktop environments available in Gentoo. The most well-known and best developed one is KDE. More recently there is also a more "light-weight" DE in development: Razor-qt. These can be complemented with applications from the desktop applications list.
Frequently Asked Questions
Please see the Qt/FAQ page.