There are a couple of new things that make this easier than it used to be:
As iamkalai posted, there's an Assignment Group metric that should be enabled by default, and it will track, for Incidents at least, each time the Assignment Group changes, and capture the start time, stop time, and calculate the duration from those. So, in a pinch, you can create a report against the out-of-the-box database view called "Incident_Metric", which is a join between Metric Instance and Incident, so that you can group by Incident, and see each Assignment Group assigned.
Possibly even easier than that, depending on your use case - if, for example, you want to see just the assignment group history for a SPECIFIC incident rather than all of them, then you can see it from the Metrics Timeline.
Assignment Group, for some reason, I don't believe is Timeline enabled out of the box, but it's as easy as navigating (or having an admin navigate) to Definitions in the Metrics application and turn on the "Timeline" checkbox. As long as the metric itself is active, then the timeline should be retro-active.
Once it's Timeline enabled, and you have Metrics collected for Assignment group, then you can click the menu at the top of a specific incident to see the history of who was assigned and when. It's also convenient for seeing which assignment group put it in a certain state - for example if you have an issue with large hold times, without the metrics, all we can see is that the ticket was on hold for 5 days and then resolved by this Assignment Group. Using Metrics and the Timeline, it's easy to see if it may have been a different assignment group that held it in a hold status for extended times before passing it to the group that resolved it.
I wish I could show better what it looks like, but the data in my instance is terrible. Here's an example from the web:
