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 ebuilds for Qt and related packages are maintained by a dedicated team. 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 a dev-qt/qt-meta ebuild for Qt4, and there are sets in the overlay for both Qt4 and Qt5.
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. There is also a younger and more "light-weight" DE: LXQt (a merge of Razor-Qt and LXDE). These can be complemented with applications from the Qt Desktop applications list.
Compiling programs that use qmake and need Qt5
Programs that need Qt5 will need its version of qmake. It is located in /usr/lib/qt5/bin and needs to be executed with its full path (i.e., /usr/lib/qt5/bin/qmake). After it's run, compilation will just work.
Alternatively, one can use dev-qt/qtchooser functionality in the development environment. Start with:
Frequently asked questions
Please see the Qt/FAQ page.