Can non-ASCII passwords somehow cause issues?

Hi, apologies if this is the wrong sub for a question like this.

Passwords are generally comprised only of ASCII symbols, so English letters, arabic numbers and symbols like ~!"#$%&, 128 symbols in total, which isn't that much. But is this mandatory?

For example could I run into problems if my password used Cyrillic letters, or weird symbols like ⅒? Could it cause issues with some websites?

ascii non passwords privacy