I remember when I lived in Australia, my friends told me that many crimes do not show up on background checks after 5 years. Also in Australia, the sex offender registry is confidential, and can only be checked for very specific things (jobs involving children, certain types of housing, etc.)
In the US, there are ways of getting records expunged for some crimes, but due to freedom of speech laws/1st amendment, typically background check companies are free to hold on to older records if they were at one time public.
I know in the EU, many countries have right to be forgotten laws, but the EFF has historically stood against them due to the fact they've often been used by wealthy individuals to hide their crimes.