Assume that you want to go to Oslo. You want to aggregate all airports that are 100 kilometres from Oslo. This includes at least Gardemoen, Rygge and some others probably.
Is there any service that tracks all flights that go there?
Similarly for any other city.
Type the airport name or code into the search box of OpenFlights, or substitute the airport's three-letter code into this link: http://openflights.org/airport/HND
Then click on any airport and the "list" icon in the popup to see who exactly flies that route.
Alternatively, look up the airport on Wikipedia, although the list of destinations is just a table.
Your first address should be the website of the respective Airport. For example, Oslo Airport has a destination map. One can in most cases assume that if there is a flight TO a location, that there is also a return flight back.
I do not think that there are services that allow you to include several airports, even less so by a km radius. You will have to go through the airports that are available for you by hand.
I use http://www.flightconnections.com/ for this. It's map based an you can click on any airport (or a near by one) and see immediately all scheduled flights including airlines and day-per-week flown.
CAVEAT: "Direct" is not the same as "Non-Stop". Many airlines now combine to more or less random legs, give it the same flight number and call it "direct". It doesn't mean it's the same plane or same crew and it certainly isn't non stop. I
Normally any flight aggregator like Skyscanner or Kayak allow searching for all airports servicing the city. In your case if you put in Oslo you will get both airports, similar behavior exists for other cities like London(LON), Moscow, Paris, New York(NYC) just to name a few.
Other services like Travelocity used to suggest additional major airports if prices were to be better or similar such as in the case of flights to Newark Philadelphia airport used to pop up.
Kayak has a direct flight search tool. As far as I can tell, it only lists departures from the airport you enter but if you click each departure there should be a "reverse direction" option which quickly links you to the departure schedule for the destination airport.
Like the other answers I'll share my tools that I use, the most common ones are: (with no affiliation to them) are skyscanner and greatescape I feel like they are both underappreciated for the amount of things they let you do.
You are able to put in the airport of your choice and then get a list or map view of all the destinations you're able to reach from there. Both allow for direct flights only to be selected, with Great Escape allowing for further filters too.
*Both of these tools were mentioned by a now deleted answer, which I presume was deleted for spam reasons, but I agree that both are useful tools so I have posted to highlight them as options
Use Google. It invented an awesome service for searching flights.
Just type in a google search - 'flights from mumbai' and it will return you a list of non-stop flights from mumbai. If you want to find all flights to Mumbai - print 'flights to mumbai'.
For anyone googling here, Google has unfortunately changed the above a little bit since this screenshot; however it's still a fairly amazing facility.
I use itasoftware.com, when you put in your city there is a button beside it that you can use to expand to airports within a certain number of miles and you can deselect the ones that are not of interest. To find closest international airport I use TravelMath.com
Google.com/flights is a good alternative to see fast where are current flight carries to one city.
I found a very similar tool, but with advanced functionality. You may see baggage allowance, aircraft model with the tail number on the route and airport schedule with arrival and departures. The synergy of the flight radar and direct flights mapping https://flightsmap.org
