This part will look at aerial military drones (and not include anything that's tethered).
The employment of aerial drones appears to follow two motives
(1) to eliminate the risk to the crew
(2) to push boundaries
The risk elimination is a relatively simple motive, and most evident in suicide drones and target drones. It's easily visible why these are enticing jobs for drones. Both were actually done by manned aviation as well, though: The famous kamikaze pilots flew suicide missions and at the very least one WW2 aircraft was armour-plated to withstand aerial gunnery training while being the target of frangible bullets (RP-63G "Pinball").
To push the boundaries of manned aviation with unmanned aviation takes several different shapes, which can be summarized as addressing the problem of maintaining a pilot's function in the aircraft.
(2a) Extreme duration
The Rutan Voyager stayed in the air with a crew for nine days on a record flight, but much less is practical in military aviation. Purposes which don't justify the employment of a strategic bomber-like aircraft will typically face a crew endurance measured in a few hours, and the associated abuse of drugs impairs decision-making.
You can pull off some missions with a slow-moving aircraft which stays in the air for a day or two, but a drone is very much favourable over a manned aircraft in this role.
The theoretical benefit of such drones is that they (seem to) make long duration flight more practical and thus make new, previously not really considered, military aviation missions feasible. This tends to be all about sensor-centric aerial surveillance and radio relay functions in practice.
(2b) Extreme altitudes
It is possible to fly very high with manned aviation - but the required pressurized cabin, the backup pressurized suit and generally the weight associated with a live crew are very troublesome, as lift is difficult to generate in thin air.
Again, unmanned aerial drones can push the boundary by offering a means to control the aircraft and its payload that's much less elaborate than required for a crew.
The theoretical benefit of such drones is that they (seem to) make extremely high altitude flight more practical and thus make new, previously not really considered, military aviation missions feasible. This tends to be all about sensor-centric aerial surveillance and radio relay functions in practice, too.
(2c) Extremely small size
There's obviously a minimum size for manned aircraft, especially if you want to employ a payload other than the pilot. Unmanned aviation can be much, much smaller - down to insect size and then there's still a camera on-board.
Small size is a huge quality in itself. It potentially allows cheap production, and thus quantity production. Imagine an army air force of 100 million bee-like drones, each equipped with enough sensory abilities, artificial intelligence and poison to identify and kill a hostile soldier. Conventional warfare would turn into a battle of tiny flying and crawling robots - or become even less 'practical' than thermonuclear war. Experience suggests that war becoming 'impractical' doesn't mean it's not going to happen, of course.
Great numbers and low prices also frustrate many known air defences; you cannot hope to defeat 5,000 € drones with 50,000 € missiles, for example. Even a 5,000 € burst from an autocannon is a questionable expense. The superior affordability of offence may defeat (oversaturate) the defence even if the defence had a 100% success rate.
Small size (and thus small weight) also allows for more intrusion into urban landscapes; drones could enter buildings through openings and inspect their interior. Small wingspan could allow aerial drones to fly under the foliage cover of woodland, instead of above; woodland could be inspected and thus be controlled much better and much quicker than ever before. This would still not help much against enemies who went underground, of course.
Small size allows for greater availability. Backpack-sized drones have been in use for years, and they provide a valuable look behind the next hill, settlement or patch of woodland. Their use is restricted by weather, red tape, effective radio range and other influences, but hopes are still high for this kind of flying eye on platoon and soon infantry or scout squad level.
Small size is appropriate for small payloads. There was a proposal for a flying sniper rifle, for example. The idea was that a RC plane would more easily be able to find, ID and engage targets with a sniper rifle (or other firearm) than a ground-based man. This and other ideas (such as simply fly a RC drone into a suspicious cloud with a chemical agent detector onboard) require only a small payload, and the minimum effort for manned aviation would be much greater than the effort required by RC planes for such a payload.
To save the mass of a pilot and his equipment (oxygen supply, seat, interfaces) may increase the payload by almost the same mass. This may yield a more cost- and logistics-efficient aircraft, which may lead to lower costs (in theory) or higher quantity.
Small size yields survivability. A bird-like drone may emulate actual birds well-enough to penetrate defences. Insect-sized drones would be too small even for a shotgun-based defence. Troops would basically need to use a mosquito net-based defence, albeit this could be defeated as well.
The theoretical benefits of pushing the boundary of aircraft down with unmanned concepts are thus greater availability, superiority through better affordability, greater survivability and greater intrusiveness. A particular design may or may not combine multiple or even all of these advantages.
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So some boundaries can be pushed and risks for air crews can be avoided. What does this really mean?
The affordability of small drones allows a return to (artificially) huge military forces. A substitution of labour by capital, except that this time it wouldn't be a substitution for many Western countries, as they lost manpower strength long ago an could now merely return to the approaches built on the assumption that millions of individuals were available for a campaign. Sometime in the future, the battalion that cordons and searches patches of woodland could actually be a truckload of bat-like drones.
The ability to keep flying for very long or to fly at extremely high altitudes - or both - approaches the capabilities known from satellites. Some small powers could emulate a national satellite support with such drones if their opponent cannot counter this. This may modernize the repertoire of a small power at war a lot, and especially so if said small power isn't backed by a great power and its satellites.
The availability of (organic!) air support down to platoon and at times down to squad level adds sophistication and combined arms abilities to units and small units (not only combat troops, but also electronic warfare, NBC and engineers, for example). An infantry or scout platoon my have more sophisticated air support and observed indirect fire support in the future than most divisions had in major wars of the 20th century. This may either introduce staff-like planning and behaviour into units and small units and ruin their command and control this way or it may introduce simplified tools and techniques to enable units and small units to make good use of drones without becoming too occupied with them. The latter scenario may even yield positive feedback to higher HQs and their handling of a sophisticated range of assets.
Small, very survivable drones which solve the energy resupply issue (solar energy, harvesting of bio fuels, or simply sitting on power lines or something hot can replenish the power reserve of a drone) could in theory reach, persist in and be effective far away from the main friendly forces; deep in hostile territory. Think about a drone equivalent of a bird or bat that flies and eats its way from your capital to theirs, and then proceeds to crawl into a transformer station, causing a blackout. Or it ignites a main kerosene tank of the hostile capital's international airport. It could also simply serve as a mobile non-permanent satellite navigation jammer.
The pushing of boundaries towards smaller size may provoke a second pushing of boundaries; the extension of mission duration and effectiveness by autonomous replenishment of onboard energy reserves.
It's fascinating how the big-ticket big drone programs such as Global Hawk succeeded in attracting so much more attention and funds than the micro air vehicles and MAV energy resupply projects in face of such prospects. Or maybe the issue is that too few people think about the theory and prospects, preferring the superficialities instead? Another explanation is that money drives publishing and generates attention, and Global Hawk et al are big ticket items, while Raven et al aren't.