Installed new cartridge. When I turn the water supply on water keeps running with the handle in the off position. Tried 3 different cartridge's. Even if you turn the water on it keeps flowing.

SOURCE: I put a new shower valve on a delta shower faucet

The first thing I would do is to start over Remove the cartridge and look behind it to make sure the Quad seal and seats and springs are all in place, It sounds like a least one of the seats fell out of their home. Also double check that the spring are in correctly with the tapered end pointing towards the cartridge. The link below may or may not be helpful.

http://www.deltafaucet.com/customersupport/documentation/1724-74.html?document=pd&documentIndex=0

SOURCE: I installed a new delta

Have you checked the seat and springs that go behind the cartridge?

SOURCE: i instslled new delta single shower faucet ,the

Sounds like someone did not remove the plastic valve assembly before soldering the valve body to the copper piping.
That or the valve body needs cleaned out. It is common for debris to get knocked lose when doing installations on old plumbing.

SOURCE: I installed a new 17

Remove handle and check temperature adjustment it maybe out. Remove handle and slide little adjustment gear out and move it a 1/4 turn and reinstall it to see if any difference occurs. If it is not set right you want be able to turn water off and possibly cartridge maybe be bad. With it being brand new and it should work out of the box take it back if no simple fixes will work. You want the best and not have to work on something you just bought anyway. Something may have been left off the cartridge maybe seat bad or o ring.