"You can make an omelette without breaking the kitchen!"
Rump kernels enable you to build the software stack you need without forcing you to reinvent the wheels. The key observation is that a software stack needs driver-like components which are conventionally tightly-knit into operating systems — even if you do not desire the limitations and infrastructure overhead of a given OS, you do need drivers.
We solve the problem by providing free, reusable, componentized, kernel quality drivers such as file systems, POSIX system calls, PCI device drivers and TCP/IP and SCSI protocol stacks. As a production-ready example, we offer the Rumprun unikernel, which clocks in at a few thousand lines of code plus rump kernel components, and supports POSIX'y software directly on both raw hardware and cloud hypervisors such as KVM and Xen.
The article Rise and Fall of the Operating System provides an extended high-level motivation for rump kernels. The book Design and Implementation of the Anykernel and Rump Kernels gives a technical description of the fundamental operating principles and terminology. Further information is available on the wiki or interactively via the community. You can also hire consultants for commercial support.