They're fun, good-handling cars, and cheap now. However, it's best to think of them as German Alfas rather than German Hondas (if that makes sense) because they can be highly-strung and suffer from niggly and potentially catastrophic faults. BMWs are nowhere near Japanese or Czech build quality.
The 2002-2004 cars with the M47N engines are the most fragile. Power was boosted to 150PS and they can blow their turbos and swallow bits of their inlet manifold swirl flaps. To prevent this, have the swirl flaps removed (dealers won't do it; independents will) and blanked using the nice little aluminium plates available from PMW, a specialist in Essex. It makes no difference to economy or performance, trust me. Forget the stupid service indicator - change the oil about every 6 months/5,000 miles and use good synthetic like Mobil 1. Also get the vacuum hoses to the EGR valve and turbo checked and replaced as necessary and check the EGR valve works, as a stuck EGR vaklve or non-functioning wastegate will kill the turbo due to excess heat. Ensure the oil separator (breather) has been changed for the current "vortex" type as the original "loo roll" type clogs and has been implicated in turbo failure. Finally, always idle the car for a minute (or more if you've just pulled off the motorway) to let the turbo cool. Post-2004 cars have redesigned swirl flaps and electronically-controlled turbo wastegates and are more robust.
Garages tend to skip air filter changes because there is a shedload of engine covers to be removed to get to the thing. Likewise, coolant changes are rarely performed. Get these done by a trusted garage as soon as you buy the car.
Niggly faults include the wiring breaking to the rear tailgate, which can affect the unlocking button, lights, heated rear window, you name it. Cheap but annoying to trace and solder back together. The rear window sealing rubbers chafe the heating elements after 100,000 miles or so and you need a new rear window. The tube to the rear washer jet always falls off and you have to remove a piece of trim and the rear spoiler to reattach it (use tape to keep it there). They can rust, unlike VAG Group cars - normally the wings and tailgate. Park distance control rarely works properly, if fitted, but can be fixed with patience and the odd new sensor. Brake pipes rust a lot (mine has a new copper one on the driver's side) and rear springs on the Touring break frequently. The rear subframe can crack on Tourings as well, although this can be fixed.
On the positive side, paint finish is excellent and the interiors are incredibly hard-wearing. Old Tourings keep a bit of credibility even after all the saloons have fallen into the hands of the barryboys and charvers. IMHO the 320d has quite enough power; the 330d is a loony machine but you'll pay for it at the pumps and it has nearly all the same potential faults.
PM me if you want to know more.