I downloaded a Ubuntu 15.04 DVD, and tried to install.

It all seems good, and gets to the "Installation type" screen, I choose full erase and install,
it goes to the "Erase Disk and start now" screen, and confirms that it will erase the full indicated disk and install.

Clicking "Install Now", then changes the window title to "Installation Type" again, but nothing else changes. Clicking "Install Now" again the window does go back to the "Installation Type" screen.

Stuck in a loop.

Choosing Quit, goes to the live version, which works, but trying an install again from there, same thing.

You might go back to the initial loading screen of the DVD and you'll see the second line of the boot screen gives you the option of installing directly.

By the way, you did check the MD5SUM, didn't you? (Or the SHA256, or whatever you used)

I did try direct install, twice.
I was only noting that quitting that (after a few trips around the loop) fell into a live session.

You don't indicate whether you verified the download by doing an md5 checksum? Are you able to boot the DVD on another machine to see if it boots to eliminate a bad burn as a possibility?

No, I didn't check the md5.
I had assumed(?!) that if it booted, and loaded, and did a full live run, it was ok.
Maybe this is wrong, and/or careless.

Does is check it's own md5 on boot/loading?

No. The link below explains how for Ubuntu.

https://help.ubuntu.com/community/HowToMD5SUM

Last I checked, there was an option in the LiveCD/DVD boot menu to check installation media. Check under "Advanced options" or such.