The water pump in any car is responsible for pushing coolant through the engine block to prevent the engine from overheating. These pumps are typically driven by the same serpentine belt that powers the alternator and power steering pump and is driven by the crankshaft pulley.
The BMW E36 shipped with a water pump featuring a plastic impeller. Though the composite BMW uses for this impeller has gotten better in recent years a common failure of the E36 cooling system was due to this plastic; it had the tendency to break apart suddenly due to age or use. This is not healthy for the cooling system and if this happens then engine overheating is imminent as coolant is not being pushed through the block to the radiator to distribute the accumulated heat. Also you'd have a bunch of plastic impeller shards in your coolant system which is not where they belong.
It's worth noting that a hot engine is not always a result of a water pump failure but a water pump failure will eventually result in engine overheating. For this reason, replacing the factory-installed water pump in the E36 with a pump using a metal impeller is a common maintainance item that supports the longevity of the engine and will protect against being stranded by an overheated engine. If the engine is allowed to overheat significantly it is very likely to sustain fatal damage requiring a full teardown. Even worse, the ECU buffers the temperature reading from the engine so the temperature displayed in the cabin thermometer lags what the engine is actually experiencing. By the time the engine overheat light comes on the engine may already be in dangerous territory.
Anyhow I performed this upgrade on my BMW 318ti. The only good thing about this car's M44 Engine being an inline-4 is that there is much more room in the engine bay. So much room in fact that the radiator does not even need to be removed to perform this maintainance as it would need to be if working on any of the inline-6 E36s like the M3. Removing the fan shroud is helpful however.
I won't detail every step of this upgrade, consult your service manual for the exact process. But I will include some insight and document my experience here.
It's a good idea to perform other maintainance items while you have the belts off and block drained. Personally I replaced both serpentine belts while they were off. A note about the belt tensioners: the secondary belt tensioner required a 8mm hex tool to untension; the tensioner for the main belt required a torx T50 bit; both of these things I did not have and needed to wait on. The secondary belt was rather easy to untension with the right tools but the main belt tensioner required a comical amount of force to move just a little bit to slide the belt off; putting it back on was a feat of strength and required a second person because I used both my hands to torque and steady the ratchet. My E36 uses a hydraulic tensioner in contrast to the mechanical tensioner used on the secondary belt.
Removing the water pump was a nightmare. I have read every forum post in existence on the topic and it seems these pumps either pop out with a little bit of extra force once all four bolts are out or are welded into the block by God and are very difficult to pull. Unfortunately mine was the latter. BMW tapped two threads on the top and bottom of the pump into which you can drive the same size bolts as are on the thermostat. In its infinite wisdom some BMW engineer also designed the pump to have a large metal fin going exactly over this tapped thread on the bottom.
I used these threads to attempt to drive the pump out. The bottom bolt got stuck as clearance between the fin and the bolt head became so small after driving it in that I couldn't fit the socket on it anymore. On the top side, the pump was so stuck in the block that driving the top bolt in broke the entire top tab of the water pump off. I seriously strained my bicep trying to pull it out directly.
This was vexing. A week after I snapped the tab we had a party at my house and I convinced everyone to take turns wailing on this thing using a pry bar after we'd been properly inebriated; nobody could make it move more than a few mm. This was probably due to only applying force to the top of the pump, causing it to lodge itself cockeyed in the block. I tried every angle over the next few days using various prybars.
Eventually I ideated a solution to this problem that would use a bearing puller to pull the pump from the block. Though the application was a bit different than what a bearing puller is designed for I figured it could work. I would drive two bolts through the holes of the bearing puller and apply equal pressure to both sides of the pump this way, mimicking what the tabs are meant to do.
The particular puller I ordered had holes tapped for an M9 bolt which is extremely rare and weird that they did this. The extensions they provided of course fit in these holes, but I needed bolts to go through these holes, not extensions to thread halfway into them. Because of their rarity M9 bolts are also quite expensive; I decided against ordering 2 bolts for $40 and looked for another solution to the bolt thing. The hardware store near me sells only the even-numbered metric bolts, unsurprisingly; I settled on using a 5/16" bolt which will fit in a 9mm sized hole and, because this would not thread in that hole on the puller, a corresponding nut on the other side of the puller to thread the bolt through. I placed two shims of wood against the block to prevent further cosmetic damage to the timing case.
This use of the puller worked and the pump popped out like it was nothing. I replaced the pump and installed new screws for the pump, as the one that got stuck in the pump tab was unable to be removed easily. I replaced the belts and, as mentioned, called on my fiancé to install the main belt while I torqued the tensioner back with strength I only cultivated over the last year.
With new coolant mixed together I refilled the radiator (making sure the drain plug and drain bolt were tightened to spec) and bled air from the system until the cabin heat ran hot when the engine was at mid-temp.
I threw the old, very scarred pump into the trash and that was that.