It depends. Versions of Unicode previous to XP did not support characters greater than FFFF. You will not be able to see any such characters on such a system. You should be able to see any new characters added to Unicode under FFFF if you have fonts available.
On XP, you may have to set your computer to see the higher characters. See http://unicode.org/help/display_problems.html .
Windows 7 has a problem with viewing some characters added since Unicode 5.1. See http://babelstone.blogspot.com/2010_05_01_archive.html for an exploration of this. Windows has announced this bug will be fixed when Service Patch 1 is released. See the last paragraph at http://www.unicode.org/mail-arch/unicode-ml/y2010-m10/0087.html .
You may only need to install new fonts to see characters in Unicode version 5.1. For general use, I suggest the free DejaVu fonts from http://dejavu-fonts.org/wiki/Main_Page , the free Stix fonts from http://www.stixfonts.org/ , the free SIL Doulos and Charis fonts from http://scripts.sil.org/cms/scripts/page.php?site_id=nrsi&id=default , and the free Symbola font from http://users.teilar.gr/~g1951d/ .
Normally, with Microsoft machines, you get updated to the most recent version of Unicode every time you install a new version of Internet Explorer. Even if you don’t use that browser, it is useful to install the most recent version to make sure your Unicode support is up to date. See http://en.wikipedia.org/wiki/Uniscribe .