When is an aircraft handed over from Center to Approach?
In general, aircraft are handed over on frequency to the next controller via a simple instruction to change the frequency:


  R: "Lufthansa One Two Tree, contact Langen Radar 128.550"
  A: "Contact 128.550, Lufthansa One Two Tree"


The simplified order of contacted stations for a commercial flight, depending on availability and airspace structure is:


  
  Delivery (if available)  
  Apron/Ground (if available)  
  Tower  
  Departure (if available, sometimes handled by Approach)  
  Enroute ACC/Center (multiple stations along the route)  
  Arrival/Approach  
  Director (if available / in use)  
  Tower  
  Ground (if available)  
  


The point and altitude or flightlevel where the handoff is initiated is agreed upon in Letters of Agreement between the different sectors or ACC/ARTCC stations. 


(Source: SentryRaven - Own Work)

The above picture shows the airspace around Köln/Bonn (EDDK) Airport, with the Köln Arrival East (KAE) sector being actively displayed in black. The STAR entry / clearance limit fixes are GULKO and KOPAG, in the east. Most traffic not arriving via NOR VOR in the west will terminate at these two fixes, which also serve as the handoff fix between the center controller (Paderborn High, Taunus and/or Rhein Radar Upper Control) and the arrival controller responsible for Köln East Sector. The Letter of Agreement between those sectors states that traffic arriving to Köln/Bonn via GULKO and KOPAG is to be cleared to FL100 and cleared for the respective STAR out of GULKO or KOPAG. The handoff is initiated approaching FL100 and approaching the entry fix, before the sector boundary. 

The handoff is therefore based on airspace structure and sector boundaries, not on a preset distance from the airport. While arrivals from the east are handed off at FL100 over GULKO and KOPAG, arrivals from the west at NOR VOR are handed off at FL140-FL170.



Landing and Take-Off clearances are issued by the Tower controller, as the approach controller is not necessarily based at the same airport but in most cases in a remote facility together with other approach and center controllers. They have no visibility of the airport and can thus not issue these clearances. 

The handoff to Tower usually happens when the aircraft is established on the final approach segment of the published approach (usually after passing the Final Approach Fix (FAF)) or a visual approach to the runway.

The phraseology is the same as above:


  R: "Germanwings Two Six Alpha, contact Tower 124.975"
  A: "124.975, Germanwings Two Six Alpha"