I recently (well, around 6 months ago) bought a Leon of the same era/chassis- thanks to the part sharing around Volkswagen Group, most of the issues I was looking out for are equally applicable to the A3. It's a bit of a brain dump but hopefully some useful information there:
ABS sensor module fault is a biggie. This has got cheaper over the years but is still a ~£400 fix. Seems to be very intermittent- some cars suffer repeated failures, some are fine and dandy on the original one. If it's not had one done it will probably be fine but worth keeping in mind.
As Va1o already mentioned, timing chain stretch on the 3.2.
The 2.0 FSI and 2.0 TFSI need new timing belts plus water pump and tensioners every ~60,000 or 4 years. This was originally 115,000 from memory but Audi/VAG issued a notice on this due to premature failures. Some sellers and a few dealers seem to think the original figure still applies- it's not worth arguing the toss with a seller who thinks this as there are plenty of other cars out there.
Maintaining Haldex can be expensive; many sellers neglect the 20,000 mile services (alternating oil/oil and filter) which leads to mechanical failures. Think these were sold as "sealed for life", like most gearboxes sold as such they should be treated as anything but in reality.
2.0 TFSI can suffer from high pressure fuel pump failures, either in the cam or in the follower. This is potentially engine destroying- the early one-piece cams were prone to failure due to a patchy manufacturing process- if one of these has survived 80,000 miles it's likely to be fine. The replacement two-piece cams were too hard and caused wear on the cam follower, leading to fluctuations in fuel pressure. Worth checking both cam and lobe for wear. The same issue may exist with the non-turbo 2.0 FSI but I don't know offhand.
Original and revision diverter valves on the TFSI can fail due to being made of perishable rubber. This causes boost leaks and can result in a CEL and error code, plus associated drop in power. The later all-metal revision of the valve doesn't suffer from these problems though some people still prefer the earlier revisions because they're a bit noisier.
Variable intake manifold tumble flaps on 2.0 FSI and TFSI can stick, this is particularly pronounced in cold weather. Usually isn't bad enough to trigger a CEL but can make for a bit of roughness or hesitancy on cold starts.
Similarly, the PCV (Positive Crankcase Ventilation) valve can die. This results in a rough idle when cold and a bit of hesitancy at low revs, plus a drop in fuel economy and small drop in boost on the turbocharged models. Easy to identify (just pull the oil dipstick out on a running engine, if it splutters it's working fine, if not it's probably dead) and quite cheap to fix too. I'm 80% sure that my Leon is suffering from this.