Are there advanced "spell checkers"?

Question

Spellchecking

As far as I know, the usual "spell checker" only checks spelling against a dictionary. I'm wondering if there are programs or online services that conduct a deeper check, (grammar, punctuation and / or spelling, taking into account the entire sentence). I understand that this task is much more complicated than just comparing with a dictionary, but there may be a check at least according to several rules.
For example:

He sit down . -> He sits down .

Ordinary spell checkers won't be able to do this, but it doesn't seem difficult to do this.
I am primarily interested in English and Russian, but also in other European languages.
I would also be happy to read something on this topic (survey so to speak).
Thanks to.