Why shouldn't I use mysql_* functions in PHP?

What are the technical reasons for why one shouldn't use `mysql_*` functions? (e.g. `mysql_query()`, `mysql_connect()` or `mysql_real_escape_string()`)? Why should I use something else even if they work on my site? If they don't work on my site, why do I get errors like > Warning: mysql_connect(): No such file or directory