Both AmigaOS 4 (final) and MorphOS will run most 3.x applications that do not directly access customs-chips/hardware, etc.
Since both OS are PPC native, yes, they do run built-in emulation layers (Petunia/JIT) to get classic Amiga apps running.
motorollin wrote: AROS
An Amiga-like operating system that will run on any PC. Will not run any Amiga software unless you use an emulator.
From the AROS homepage:
"AROS is a portable and free desktop operating system aiming at being compatible with AmigaOS 3.1, while improving on it in many areas."
"How compatible is AROS with AmigaOS?
Very compatible. We expect that AROS will run existing software on the Amiga without problems. On other hardware, the existing software must be recompiled. We will offer a preprocessor which you can use on your code which will change any code that might break with AROS and/or warn you about such code."
"What hardware architectures is AROS available for?
Currently AROS is available in a quite usable state as native and hosted (under Linux, FreeBSD and NetBSD) for the i386 architecture (ie. IBM PC AT compatible clones) and hosted (Linux and NetBSD) for the m68k architecture (eg. the Amiga, Atari and Macintosh). There are ports under way at varying degrees of completeness to SUN SPARC (hosted under Solaris) and Palm compatible handhelds (native)."