The water pump is a very sturdy unit, but has been known to wear out. It will make a wobbling sound if the bearings are bad, and will shortly start leaking. The only place a DeLorean water pump will leak is from the weep hole that is immediately behind the pulley at six o'clock, i.e., pointed down. You will need either to lie down under the rear of the car with a flashlight and look straight up at the water pump or get a mirror on a stick to see it.  
Any other leak from the water pump area will be a split hose or loose clamp on one of the four hoses that are associated with the water pump, or the two large hoses on the left and right sides of the water pump. The coolant hoses routed under the intake manifold are inaccessible without removing the manifold itself. If coolant is running from underneath the water pump, between the cylinder banks and toward the crankshaft pulley, it is most likely coming from one of the two hoses that connect to the back of the pump. The lesson is to replace these hoses and clamps (included with our water pump kits) and tighten these hose clamps fully when replacing the water pump. You don't want to go back.
The water pump is sold as a complete Water Pump Assembly PN K102119 which includes all the clamps, hoses, gaskets, intake gaskets, and hardware required to replace it. The pump is supplied with a new steel pulley already installed, so you don't need to deal with swapping the pulley.
The pump is also sold as  Water Pump Assembly (Partial) PN K102119A. This is the same kit as above but is intended for installation at the same time as a Complete Cooling Hose Kit, and therefore omits the installation gaskets and small hoses to avoid duplication as those parts are included with the Complete Cooling Hose Kit.
Replacing the pump is a fairly involved process. Here are the basic steps:
1 - Disconnect fuel lines at the fuel distributor, the throttle linkage (remove the three bolts holding the linkage to the intake/water pump, and the ball stud link underneath the spool), vacuum hoses, and the four 7mm (10mm head) bolts holding the manifold in place. Lift the manifold off the engine and set it aside. You may need to remove the W-pipe and/or the air metering assembly for access, but generally these can be left in place. 
2 - This is a good time to verify the source of the leak if desired. At this point you can still pressurize the system. 
3 - Remove at least the block drain above the oil filter on the passenger side. Even better, also remove the drain above the catalytic converter (harder to reach). These plugs use the same 8mm square tool that is used on the oil pan drain. 
4 - Remove three bolts holding the pump to the engine. Remove the two side hoses and all the rear hose clamps. Remove the belt and pull the pump off the engine away from the hoses. If you didn't drain the block this will fill up the engine valley with coolant at this point. Remove the thermostat and back cover for re-use. New bolts are supplied with the kit, as are new gaskets for the back cover and thermostat. 
5 - Clean out the engine valley, paying special attention to the 4 very deep casting voids in the center. This is important for the longevity of the engine block as corrosion in this area has proven to be a problem. You will usually find the engine valley to be full of dirt, nests, bugs, and previously lost parts. 
6 -  Install the back cover, thermostat/cover, and plug fittings to the new pump after cleaning up the mating surfaces. Use a good sealant on the back cover gasket. 
7- OPTION - The two O-rings under the Y-pipe are included in the kit. We recommend that the Y-pipe be removed and resurfaced as part of the water pump replacement. CAUTION there are four 7mm bolts that hold the Y-pipe to the engine block. They are highly prone to breaking off! Do not attempt this part of the job unless you are familiar with bolt extraction/rethreading (heli-coil, timesert) processes. If the pipe has been leaking at that joint you will need to find competent assistance.
8- Clean up the Y-pipe and heater tube, and install the new back hoses. Install the new pump assembly to the engine. You may need to trim the two back hoses slightly to the correct length. Tighten the hose clamps securely, wait a while and tighten them again! Install the new side hoses and connect the large hoses at the thermostat and the body of the pump. New hoses are recommended here if they don't seem "fresh." Install and tension the belt at the alternator
9 - OPTION - In case you haven't noticed, this would be a great time to change the  spark plugs and distributor cap/rotor/wires. Also a good time to upgrade the fuel injection lines to new metal-covered fuel injection lines. Those plastic fuel lines are well over 35 years old at this point. Access will never be better. 
10 - Fill the system with coolant. Bleed the thermostat and the top of the radiator, and check for leaks. If you have a pressure tester, pressurize the system and check for leaks on the hoses you just installed before you close it back up. 
11 - Reinstall the intake manifold. Connect the fuel lines and hoses. Recheck all your connections again. Start the engine and immediately inspect the fuel lines for leaks. Once you are satisfied that the fuel system is tight, run the engine long enough that the cooling fans cycle on and off several times. Monitor the temperature gauge during this process, as trapped air in the radiator or engine can cause an air lock and overheat the engine. If the gauge gets to the 220 degree mark, stop the engine immediately, allow it to cool, and re-bleed the thermostat and the radiator. 

By Dave Swingle, 6/9/16
Revised 8/14/2016 clarified pump R&R steps.
Revised 1/13/2022 ET (formatting and links)