By far the most common method for in-flight refuelling is the
"probe-and-drogue" system, in which the tanker unreels a hose behind it
with a drogue on the end (a meshwork cone whose drag keeps the end of the
hose in a stable position).  The receiving aircraft has a probe attached to
it, which is inserted into the drogue to link the fuel systems.  Some
receiving aircraft have probes permanently mounted, some have bolt-on
probes that can be attached if a mission requires them, and some have
retractable probes.  This method is used by the US Navy, modern Russian
aircraft, and every other country that uses in-flight refuelling.

The US Air Force alone uses the "flying-boom" system.  In this system, a
rigid boom, with control surfaces on the end, is extended from the tanker
and inserted into a socket on the receiving aircraft.  This method has two
major disadvantages over the probe-and-drogue method.  First, the boom has
to be attached directly to the tanker's fuselage, which prevents refuelling
from detachable pods attached to a tanker's wings (allowing more than one
receiver to link up at a time) or to the centreline hardpoint on a fighter
or strike aircraft (allowing such aircraft to refuel each other without a
dedicated tanker), both of which are commonly done with probe-and-drogue
refuelling.  Second, the equipment on the receiving aircraft is
incompatible with the probe-and-drogue system, which means that USAF
aircraft can neither refuel nor receive fuel from any other aircraft,
including the US Navy's.

The reason why the USAF puts up with this is that the flying-boom system
can achieve much greater fuel flow rates than the probe-and-drogue system
(mainly because the rigid boom is shorter and wider than the flexible
hose).  This is mainly for the benefit of large bombers such as the B-52
and B-1; refuelling such large aircraft by probe-and-drogue would take much
longer, enough (in the USAF's judgement) to cause significant tactical
problems.  Few other air forces operate aircraft of similar size; the
handful that do are prepared to live with the refuelling delays in the
interests of compatibility.

The third refuelling system used is the "wingtip-to-wingtip" system, used
only by older Russian bombers.  In this system, a hose is unreeled from one
wingtip of the tanker, and caught by a socket in the opposite wingtip of
the receiver; the two aircraft then fly side by side, with the hose joining
their wingtips (the length of the hose is comparable to the wingspan of the
aircraft).  The tankers are all converted bombers themselves, mainly the
Myasishchyev M-3MS-2 "Bison-B".  This system is very tricky to link up,
occasionally dangerous, only usable with bombers (smaller aircraft can't
carry the necessary receiving equipment on their wingtips), and gives flow
rates even worse than probe-and-drogue; not surprisingly, the Russians have
largely replaced it with the probe-and-drogue system, and it will probably
become extinct with the retirement of the last M-3 tankers in 1994 or 1995.