The idea of using a smartphone as a full desktop PC when connected to an external monitor isn't new. The now-dead Ubuntu for phones and tablets had 'Convergence' as one of its main features, allowing you to run full desktop Linux apps when you connected a keyboard and mouse (and display, if your device supported that).
There were a few similar concepts, but the first mainstream implementation was Samsung DeX. Once you docked a Galaxy S8 (and later, Note8), you could use Android apps on a larger monitor. While still a niche solution, it received generally favorable reviews, and Samsung has continued to polish the experience since then.
Samsung today announced 'Linux on Galaxy,' which will allow users to run desktop Linux software on their Galaxy phone and DeX. It will be able to "run multiple operating systems," presumably meaning different distros (like Ubuntu, Debian, Arch, etc). The signup page says that Linux on Galaxy uses "the same Linux kernel that powers the Android OS," so there's no emulation or virtualization involved. Presumably, this uses a chroot, similar to crouton on Chrome OS and the various root-only Linux installer apps for Android. Put simply, performance should be very good.
Since there is no emulation or virtualization, it is very likely that only Linux software compiled for ARM will work with Linux on Galaxy. While Linux software is usually compiled for multiple architectures, there is some x86-only software, like Steam. Samsung didn't mention specifically which devices were compatible, but I imagine the S8/S8+ and Note8 are included.
Samsung says that Linux on Galaxy is still in development, but you can sign up for notifications of availability here. If this works as well as Samsung claims, it could make DeX a far more interesting product, and even more of a desktop-replacement for some users.
Samsung has uploaded a video showing a demonstration of Linux on Galaxy. It shows various popular Linux programs installed on DeX, including Firefox, Thunderbird, Terminal, Eclipse, and even GIMP: