So, how do you manage API versioning when your schemas are always changing?

We're running into some issues with our main APIs because the data models underneath keep changing. We've looked at a few ways to handle versions, like putting the version in the URI or using headers, but I'm wondering what's worked well for other people. I'm especially interested in solutions for when the schema changes a lot. What do you think are the best ways to manage API versions and keep things compatible for older stuff without making a ton of extra work?