Missed the May 31st deadline now what?

Anyone that can tell me a little information about what will happen to me if I missed the deadline. Can you still file and pay a fine or do I have to wait for another date?
Thanks in advance for some helpful advice

File as soon as you can. The longer you wait, the heavier will be the fine.

But I believe the fine is simply the interest on the money you owe in taxes.

Too bad they do not pay you interest if you have a tax refund coming to you (that takes 3 months)