Roundabouts (or traffic circles, and I understand they're commonly known as in the US), are extremely common here in Australia and many European countries.

I think they are generally a "danger zone" for cyclists in my opinion, particularly the large multi-lane ones.

First there are the approaches. Many cars think they should race you to get to the entrance line on the roundabout, which is often slightly narrower than the approach road, giving an ideal opportunity to be squeezed off the road. You have to “take the lane” to avoid being overtaken.

Because of the “give way to the right rule” (same in all countries I think), many cars will try to race across if they think they have time to beat another driver to their right who is just about the enter the roundabout (or who already on it, but some distance away). This creates a bad driver mindset for a cyclist who might be in front of the driver, because the driver is looking to their right, and not directly ahead - increasing the risk of being rear-ended.

Some drivers just won’t see you when you are crossing, so having some kind of lateral hi-vis (tire rims etc.) is really helpful. This is much worse at night.

Multi-lane roundabout are the worst if you have to negotiate multiple exits. If you are going 3/4 of the way round, you need to position yourself in the right hand lane, and on a big roundabout (some European ones can be over 100m in diameter) you are sitting in the middle of 2 or more lines of relatively fast traffic with drivers who are turning and searching for their exit. Not good….

At busy times, I will simply avoid these, and makes multiple pedestrian crossings to work my way around the circle.