So many different answers. Instead of following them all, I created a new project. As simple as possible; if an IntelliJ update breaks my project, I want fewer unused features to be migrated in 2018
I was getting
java.lang.ClassNotFoundException: com.badlogic.gdx.ApplicationListener...
Caused by: java.lang.ClassNotFoundException: com.badlogic.gdx.ApplicationListener...
After upgrading IntelliJ from 2017.3 to 2018.1.
This time I did not check the opposite:

and similarly for the "Advanced" settings:

Without choosing IDEA, the project took 10 seconds to generate and various instructions were given on how to import, which was noticeably more successful (the old ways were best remembered by someone Eclipse?).
Of course, I got a few more errors, but nothing that I had not encountered before, getting a new project from scratch. After that, you just need to copy the old files. Another tip, no matter what settings IntelliJ offers to open, to set the Java level to 1.7, the Gradle files do not touch, and therefore you must hunt for the remaining 1.6 links manually.
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