The original DeLorean radiator was a two-core, brass core unit with plastic side tanks that was manufactured in Belgium. It was an extremely efficient and durable unit with one weakness - the plastic side tanks. The plastic could become brittle after many years and would not stand rough or impatient treatment when changing the hoses. Many mechanics have had the experience of the neck coming off in their hand before the old hose broke loose from the plastic.
 An even more common problem is that the tanks may become unfastened from the body of the radiator. The metal fingers that hold the tanks can loosen with time and leak at the joint between the plastic and the brass. There are still many of these radiators in service today. A popular upgrade in the past was a 3-core brass radiator with metal tanks. This worked well enough once it was in place. Although it had 3 cores, there were fewer horizontal tubes per rows, so the overall efficiency of the radiator was not improved over the original. Fitting the unit in place was problematic due to its added thickness.
Avoid trying to remove the small "cap" on the driver side of the radiator. If it's leaking the seal is compromised, and attempting to remove or tighten it will often just shatter the tank.
DMC sells an American-made two-core, all aluminum, high-efficiency Radiator PN 100486 with welded aluminum tanks. This radiator has the same mounting arrangement as the original but has an added drain valve for convenience. It also eliminates the leak-prone upper radiator cap. You may find it necessary to slightly file the corners of the fan shroud where it meets the radiator as fit will vary slightly from unit to unit due to welding variations. When installing the fan shroud be sure to inspect the clearance at the corners before tightening the attachment bolts. 
When ordering a radiator, it's recommended that you also get the associated 5 mm hardware that holds the assembly together as the original hardware often breaks. This is shown as items 1, 12, 13, 14, 23 on the main radiator parts page
This is also a good time to replace the lower Rock Screen PN 101628. This part is often rusted, the lower studs tend to break on removal, and you have to detach most of it during the process of changing the radiator. Be sure to inspect the fan mountings, shroud, and radiator cowl for broken or cracked parts as you will want to replace the broken parts at the same time as the radiator. 
By Warren Wallingford, DeLorean Motor Company (Texas) with edits by James Espey
Updated 8/14/2016 DAS [details, added parts links][Clarified row vs core]
Revised 1/13/2022 ET (formatting and links)