South African Airways flights to 'less developed' locations in Africa often carry a mechanic and some spares on board.
If there are no suitable repair facilities, supplies or maintenance personnel available at the destination, then their own guy can fix any minor mechanical issues.
If there's a major problem then they would have to fly in additional repair supplies and staff (and fly out the stranded passengers).
I found this out on a flight from Johannesberg to Lilongwe (Malawi).
A seat was broken in the passenger cabin and SAA's on-board mechanic was able to fix it after a short delay.
The captain explained briefly why he was on board.
I assume that other airlines or operators who operate in undeveloped areas would do the same.
It's a lot cheaper to carry a mechanic and a few spares than it is to have an aircraft, crew and passengers stranded.
If anything on the MEL is broken then an airliner isn't airworthy so even a 'minor' issue can ground it.
I guess that the military might do the same for similar reasons, but I have no idea at all about that.
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All of the 747-100/200 freighters I flew in the 1990s had FAKs (Fly Away Kits) aboard.
I just checked a couple of the weight & balance data sets I still have from having done weight & balance work for one of them up until 2013, and the weights were 1967 and 1939 lbs.
If the aircraft was a nose loader, the FAK was usually along the side of the in the vicinity of the retractable ladder used to get from the main deck to the upper deck.
If the aircraft was not a nose loader, the FAK was always up against the main deck forward bulkhead.
The FAKs were constructed of heavy duty plywood and opened from the top.
One of the airlines had two BOW configurations for each aircraft, one with its FAK, one without, to expedite weight and balance calculations with or without its FAK.
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I've heard what you're describing called a "Fly Away Kit" or FAK.
After some brief googling, it seems the exact contents vary depending on the aircraft, operator, and the manufacturer of the FAK, but most seem to contain tools, spare tires, and other items for simple repairs.
Smaller kits are designed to fit behind a panel in the main cabin, while some larger ones are designed to fit in a Unit Load Device in the cargo hold.
I found a brief description here .
A fly away kit is a bunch of small spare parts and some tools and fluids to get the plane out of a place you don't have any support for the airplane.
We have one on all of our 727's because we operate into remote places that would take forever and a day to get a simple part to us that is a show stopper for the flight, it allows us to swap the part and fly away.
Things like spare landing light bulbs, radar altimeters, instruments, gyros, gauges and things like that are generally what I have seen and used from in there.
In addition we usually have a spare tire and sometimes a brake assembly somewhere in the belly.
Example of kit from airliners.net :
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Yes.
If a US-registered aircraft is to be operated at night, it must carry: §91.205
(c)(6)
One spare set of fuses, or three spare fuses of each kind required, that are accessible to the pilot in flight.
Now, as far as other types of spares go - no clue!
But some spares must be carried if certain requirements are met - eg, your airplane has fuses accessible from the cockpit.
In the event of an electrical problem, the flight crew can replace the fuse as required.
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When I flew C-130 transporters in the RAF, we always carried a FAK and a 'Ground Engineer', but then the extra weight was of little consequence compared to the delay of the mission in some remote location.
I now fly Business Jets, where there is significantly less room, less available (spare) payload weight, and certainly no-one qualified to make the repairs on board, so the only thing carried is a small set of wheel chocks, a couple of cans of oil and clamps for the trust-reversers.
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Normally planes don't have major spare parts.
Instead of parts they have redundancy in almost every flight system, which under certain circumstances an experienced pilot can take advantage of in order to keep operating the plane safely.
Have in mind that the weight issue in a plane is very important.
That's why a popular phrase between pilots says: Every takeoff is optional, but landing is mandatory
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I was curious to know, are there (or were there) any aircraft that regularly carry spare parts to be used for replacement or repairing the aircraft itself? If so, what are those spare parts?  I'm not talking about built-in redundant systems like a second radio, but rather actual physical spare parts. For example, a car carries a spare tyre.
Do any aircraft carry spare parts for making repairs?
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<commercial-aviation><aircraft-maintenance>
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Planes do not normally carry major spare parts. However, South African airlines carries spares to "less developed" locations in Africa. RAF planes and transporters in the 1990s carried a "Fly Away Kit". Nowadats there is little room for anything major. Major spare parts cannot usually be carried on board due to space and weight requirements. However, planes do tend to have what is known as a "fly away kit". There are some instances in which there will also be an engineer or mechanic on board.
Planes do not usually have major spare parts but they do tend to have what is known as a "fly away kit". There are some instances in which there will also be an engineer or mechanic on board. Major spare parts cannot be carried on board due to space and weight requirements.