Today, the leading derivatives for other hardware platforms is Mythbuntu. An official Ubuntu derivative, Mythbuntu is designed for MythTV, as its name suggests.
Distributions with a small installation footprint are popular for use with older or embedded systems, or among those for whom speed in computing is everything.
One current small installation is the Ubuntu Mini Remix, a Live CD whose image is only 165 megabytes -- or "only the minimal set of software to make the system work," according to the home page. The current release includes a 64 bit version, but apparently only a 32 bit version will be available in future releases.
However, perhaps the ultimate compact installation is Damn Small Linux, or DSL as it is often called. DSL installs to only 50 megabytes, and is supposed to be installable to a 486 machine with 16 megabytes of RAM (assuming any such machines are still in operation). In the interests of saving space, it includes many applications not generally seen in most distros, such as the Dillo web browser and the Ted word processor.
Some derivatives are mainly localizations -- that is, translations into other languages. For instance, you can use Baltix if you speak Lithuanian, Latvian, Polish, Norwegian, or any other language spoken around the Baltic Sea, or BOSS if you speak one of the languages of India.
Spain is particularly wealthy in derivatives, with LinEx in Extremadura, Molinux in Castile-La Mancha, and LliureX in Valencia. So is South America, particularly Brazil, whose derivatives include Tucunare, Satux, Big Linux, and Poseidon.
Many of these localizations have other purposes as well. Poseidon, for example, bills itself as "the scientific GNU/Linux," while Tucunare is intended to help improve citizen involvement with government and social concerns. In the same way, many Spanish derivatives are connected to projects concerned with self-government and increasing the availability of computers in impoverished areas.
Derivatives are particularly popular for security and privacy. Many derivatives in this category are available on Live Media, because some commands need to be run with hard drive partitions unmounted, or isolated from any possible intrusion.
Among the derivatives in this category are CAINE, which specializes in computer forensics; BackTrack, a highly customizable distribution for penetration test; and Ubuntu Privacy Remix, whose goal is to provide an isolated working environment for material that you wish to protect.
The current multimedia derivatives tend to be based on Ubuntu and are often Live CD/DVDs. The best-known is Ubuntu Studio, which includes audio, graphics, and video tools, and pays special attention to support for Wacom drawing tablets and similar hardware. Other all-round multimedia derivative is Artist X, eAR OS and APODIO.
Others are more specialized. For example, the still-in-beta PC/OS includes support for a full-array of multimedia codecs, many of which are proprietary or have a questionable legal status on Linux.
This is not an exhaustive list of Debian and Ubuntu derivatives by any means. With more space, I might say something about SymphonyOS, a radically different desktop that is no longer in development, or Ichthux and ChristianUbuntu for Christians and Sabily for Moslems.
For those who are interested in migrating a business to Debian, there are CoreBiz and Univention Corporate Server; for education, there is the long-established SkoleLinux and Edubuntu, and for the blind or vision-impaired, Oralux, which installs and runs via speech synthesis.
Its not quite true to say that, if you can think of a specialty, then there is at least one Debian or Ubuntu derivative is designed for it. However, it is not much of an exaggeration, either. Through their hosts of derivatives, Debian and Ubuntu have an influence even greater than either has alone, or even have together. If you can't find a derivative distribution in a particular category based on Debian or Ubuntu, chances are that you won't one based on any other distribution, either.
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