This article is from the comp.sys.acorn FAQ, by Paul Vigay with numerous contributions by others.
This is a fairly brief introduction to the issue. If your question isn't answered below than it is well worthwhile going to the Acorn Emulation Pages at http://acorn.cybervillage.co.uk/emulation/, which cover this subject in some detail.
PC - The early 'Archimedes' machines can handle 720K DOS discs and all machines since the A5000 can also handle 1.44MB discs. In RISC OS since version 3.0, DOS compatibility is built in, whereas under the older RISC OS 2 you have to use one of the (PD) utilities. There are two PC (software) emulators that can handle most PC software, three if you count the fact that Acorn's offering is split into two emulator programs. There are also PC cards, containing an 80x86 processor and other PC hardware, which uses the main computers' disc and video. All emulators are multitasking (and not PD). RiscPC machines, by adding a 486 or 586 'second processor' card, can fully emulate a PC. However, these are still quite slow by todays PC standards.
Unix - Unix software can be ported (and in fact many packages already have been) with the help of UnixLib. Memory-hungry ports, such as gcc, can be run with the aid of !Virtual (also PD). Of course if you are really desperate to run Unix software it might be advisable to buy RISCiX (only available second hand now), the Acorn flavour of Unix (suitable only for A540 or older machines), or perhaps use linux (compatible with all 32bit Acorn machines) or NetBSD/arm32 (for RiscPC machines).
Further information on NetBSD/arm32 is available from http://www.netbsd.org/Ports/arm32/
Mac - Commercial software is available that can read HFS format discs, including hard drives and CDROMs. For floppy discs this generally encompasses high density discs (1.44 MB format) and a limited subset of double density (720 kb) discs. Essentially the software can read the disc if the tracks have been laid down with constant angular velocity, rather than constant linear velocity - which requires a variable speed drive to work. Most Macintoshes default to CLV format discs when formatting double density discs but some Mac software exists to override this and, in theory, such discs are readable using the Acorn software. To be sure that the double density disc is a CAV format disc, then formatting it first in your Acorn machine is a good step and the disc will then work with Macintosh machines. Also most modern Macintoshes can, like Acorn machines, read and write PC format discs so transfer is possible via that medium as well. There is no Mac emulator available natively. However if you have a PC card in your RiscPC it is possible to run Executor 2.0 under the PC Emulation to then emulate a Mac. However this approach is likely to be fairly slow.
A shareware application called ExMac is available from http://www.vigay.com/cgi-bin/ROlinks?q=exmac which will allow you to read, write and format Apple Mac discs under RISC OS.
Atari ST - Like PC floppies, the Archimedes can read, write and format ST-format floppies. An Atari emulator is available, called STem, which is of limited utility is currently under development and improvement.
Amiga - The amiga uses an unusual disc format that is not easily read by other machines. Accordingly there is currently no Amiga format disc reader available. But Amiga's can read PC and Macintosh format discs, so again transfers can occur via that medium. There is one Amiga emulator available, a port of the Unix Amiga Emulator, but it is somewhat CPU intensive and really requires a StrongARM to be useful.
Spectrum - There are emulators around for this and Amstrad, the owners of the Sinclair copyright, have released permission for the ROM images, needed to run these emulators, to be copied and released with the emulators.
Apple][ - Again emulators are available for this. However they, like the Spectrum emulators, require a copy of the ROM image to work. The copyright of the ROM image for these machines, as far as I am aware, prevents them being distributed, so you have to source your own copy of them.
BBC B - Once again emulators exist, including Acorn's own effort of 6502Host. The emulators are fairly good, offering a high level of compatibility. See question 8.3 for more details about two commercially available emulators. Also various programs do exist that allow the newer Acorn machines to read BBC B format floppy discs.
GameBoy - An emulator exists for this, capable of loading and using most snapshots with sound correctly emulated. Snapshots seem to be fairly freely available on the Internet and a search in any of the more capable search engines should quickly turn up a set of sites with downloadable snapshot images.
Amstrad CPC - Emulators exist for these machines and Amstrad & Locomotive Software have given their permission for the ROM images, needed to make them work, to be freely distributable with the emulators.