There are currently 107,156 graphic characters in Unicode.
OpenType fonts are currently limited to 65,536 glyphs. Do the math. You need at least two fonts to include all Unicode characters.
Arial Unicode is an old Microsoft font. The most recent versions cover all glyphs in Unicode version 2.1, but Unicode is now up to version 5.2. See http://en.wikipedia.org/wiki/Arial_Unicode_MS . The most recent Arial Unicode contains only 38,917 characters.
James Kass has for years been providing fonts to cover all of Unicode for a very low price; See http://www.code2000.net/ . These fonts are rather ugly in some of the characters, don’t have bold or italic, but at least you have three fonts which will show something reasonable beyond an empty square. A difficulty is that if you have these fonts installed, because the fonts have so many characters, font substitution in browsers is liable to pick up less desirable forms of the characters from these fonts instead of more attractive shapes from other fonts.
Perhaps best is to accept anything for languages where you don’t understand a single letter in any case and concentrate on languages you use and on mathematical and technical symbols to appear in attractive fonts.
For Latin (and in some, Greek, and Cyrillic letters), including many unusual ones, I recommend:
the free DejaVu fonts from http://dejavu-fonts.org/wiki/index.php?title=Main_Page ,
the free Doulos SIL fonts from http://scripts.sil.org/cms/scripts/page.php?site_id=nrsi&item_id=DoulosSIL_download#FontsDownload ,
the free Charis SIL fonts from http://scripts.sil.org/cms/scripts/page.php?site_id=nrsi&id=CharisSIL_download ,
the free Junicode fonts from http://sourceforge.net/projects/junicode/files/ ,
the free Palemonas MUFI fonts from http://www.mufi.info/fonts/#PalemonasMUFI , and
the free Quivira font (no bold or italic but lots of math characters) from http://www.quivira-font.com/downloads.php .
For symbols there is the Symbola font from http://users.teilar.gr/~g1951d/ along with a lot of dead language fonts and some very nice Greek fonts.
If you want a free or inexpensive font for a particular script, a good place to look is http://www.alanwood.net/unicode/fonts.html .
For web browsers, I find Firefox does a much better job at font substitution compared to other browsers. Usually, if even one font on my system has the requested character, it will show it where other browsers are more likely to fail.