There are 16 of these planes. This first block of 65,536 characters is what you can encode with only two bytes (e.g. UTF-16), and it includes most of what anyone alive needs to encode their languages adequately enough. For a long while anything encoded beyond this block had only limited support, and plenty of bugs and limitations meant that using it was tricky (well, it worked fine in LaTeX of course; via xelatex for example). This was back in 2008/2009.
Characters encoded beyond the BMP in plane 1 and 2 included things like esoteric CJKV additions (East Asian ideographs) not usually in daily use, but part of historic documents.
Then came the emoji additions (a core set is part of the BMP and came from Japanese telecom standards), and support is now ubiquitous. Using UTF-8 is a no-brainer for most applications, and a good things that is too!