Do countries with universal health care also provide dental coverage in that way?

Where I live in the US, dental costs have been reasonable so far. But I've heard of some people getting sacked with bills in the tens of thousands, so I ask for their sake and others who could get so 'hit'.
--------------------
Here in the UK dental work is free for the under-18s, the elderly and people on low incomes. The rest of us have to pay a contribution to our dental fees but it is still heavily subsidised by our universal health care system (the NHS). Ive never had to pay more than about £15 (about US $25).
Source