The UK doesn't allow it. In fact strictly speaking it does in that you don't need ATC at the airport, but you do need a "controller" controlling the inbounds. An example is Walney Island EGNL where the man in the tower is a FISO not an ATCO, and the approach controller (who has to be an ATCO) is located remotely. There are other examples but they involve an operator (e.g. an air ambulance) with a special permission, who is flying an "unpublished" IAP (which may be his intellectual property) into a FISO or A/G staffed airfield.
The remote controller does not need to "see" any traffic; he can work totally procedurally so he could be a man in a hut 10000 miles away, so long as he has the VHF radio. But he has to be an ATCO which means a specific pay grade.
In the USA it's the same AFAIK. There needs to be an approach controller "somewhere".
I know that this restriction is loosening somewhere in Europe.
In Scotland there are a couple of airfields that may be moving towards this. I have read that GPS/RNAV approaches have been approved at Barra and Benbecula which have only a FISO. What I don't know however is whether these will be required to use an approach controller located somewhere; if they do then they are no different to the long standing Walney Island case and it is nothing new at all.
The key to this, in the UK at least, is who is going to pay the ATCO desk cost (or cost share) which is substantial (6 figures) even for day-only cover. You could have a GPS approach at any FISO airfield if you are willing to pay for that 
And the obvious workaround - using a FISO to schedule the traffic - will drive a coach through ATC pay grades which is simply not an option.
Germany doesn't even allow VFR approaches without local AFIS...
Australia doesn't require one - i have done plenty of NDBs into Bathurst with controller at all. In the US you are talking to a radar controller before going over to the Unicom.
As a practical matter, in the US there is usually approach controller near the larger airports or an enroute center controller that clears the aircraft for the approach procedure. However, in many areas, there isn't radar or radio coverage in the vicinity of the airport, so the controller may have to issue the clearance some distance from the IAF. With RNAV (GPS) approaches at most of the outlying airports, many are designed with what is called a TAA (Terminal Arrival Area). The TAA is usually in the form of a T design with IAF's at the ends of the T and at the center of the T. An area that arcs thirty miles from each of the IAF's, one for straight in's to the center IAF and one each for left base and right base entry area, permit approaches without the need for procedure turns and without the involvement of controllers to vector the aircraft once inside the TAA boundaries. When an aircraft is cleared for the approach, on entering the TAA sector, they are expected to automatically descend to the charted altitude for the sector, proceed directly to the IAF and commence the approach without needing ATC involvement. There is also a special clearance called "Cruise" which permits the aircraft to navigate to the airport, and all altitudes below the cruising altitude are clear of IFR traffic. The pilot is authorized to descend at their discretion along the route or airway down to the MEA, but once an altitude is vacated, they may not climb back to the altitude without a new clearance. When they reach the vicinity of the airport, the pilot can use any authorized feeder route to any approach available at the airport and conduct the approach, all without ATC involvement after having received the "cruise" clearance.
@achimha, do you think that will change under SERA?
Never ever. As you will know, it's a holy holy cow in Germany. However, to be precise, it's not an AFIS which is required to operate at german airfields; a mere A/G operator is sufficient.
And re this topic specifically, yes, you can fly IFR approaches without ATC in Germany, as long as there is an operative AFIS at the field.
So, the FISO has the authority to schedule traffic for the approach procedure?
So, the FISO has the authority to schedule traffic for the approach procedure?
No. At some point you leave controlled airspace (F in Germany but same as G in UK) and ATC will tell you "radar service terminated, contact Schwäbisch Hall Info on 129.225". You then call AFIS "Schwäbisch Hall Info, D-EAAA ILS 28 5 DME" and you will get a "D-EAAA, Schwäbisch Hall Info, roger, no known traffic at this point, landing at your own discretion".
do you think that will change under SERA?
No. SERA is about the rules of the air and is directed at ATC whereas the German ground operator requirement is an airfield regulation. Airfields in Germany are under supervision of the Bundesländer and some of them allowed unmanned traffic but some years ago they all sat together to find a common position and unfortunately the unmanned operation is gone. The "reason" is assistance in case of an emergency. You need to have a person present that knows how to operate the emergency equipment and alert the fire brigade and whatever. Can be your neighbor who drives to the airfield but practically speaking airfields are very inflexible in handling this. The aeroclub airfields are the most flexible ones.
At some point you leave controlled airspace (F in Germany but same as G in UK) and ATC will tell you "radar service terminated, contact Schwäbisch Hall Info on 129.225".
Presumably you are supposed to be VFR at that point i.e. your IFR clearance has been terminated implicitly.
I am wondering to what extent the often stated requirement for an approach qualified ATCO to be around to schedule traffic for an approach procedure is an ICAO requirement.
The very high cost of ATC (or the very high cost of the alternative i.e. paying a nearby ATC unit to provide an approach service) does block the introduction of GPS approaches to the vast majority of GA airfields in the UK. None of the airfields which are currently non-ATC will be able to afford it.
I have read that GPS/RNAV approaches have been approved at Barra and Benbecula
If they have, they're not in the AIP yet.... The other thing is that there are quite frequent radar and GPS jamming exercises by the military In the Highlands and Islands which would render any GPS approaches unuseable for those periods....the next one is scheduled for this Monday...
