When does it make sense to add a message queue to a system?

I've been thinking about when message queues really make a difference. I mean, past just simple async stuff. When do they really start to look good? Is it mainly about separating services, dealing with big traffic spikes, or helping build things like event-driven systems? I'd love to hear about actual times where a message queue was obviously the right choice, not just something extra. So, what are the big signs that scream, 'Hey, time to add a message queue!' to your setup?