Flight is one of the most recognizable abilities of typical dragons in fantasy and mythology. Most times, it's what divides dragons from serpents.
But in real life flight in typical western Dragons seems highly unlikely. This is mainly because of two reasons:
- Dragons are vertebrates, and as such, they cannot have four legs and two wings, therefore six limbs
- Usually, dragons are portrayed as very big, and animals that big couldn't fly
Let's look at these characteristics in detail:
Number of limbs
Tetrapods usually have four limbs. Those evolved from the fins of early Tetrapodomorpha when they gained the ability to walk an land. Some tetrapods, such as snakes, have lost those limbs, but there is no known species of tetrapod that has more than four legs.
However, there are cases of tetrapods having more than four limbs. This condition is called polymelia and is considered a deformity. For example, there are chicks which are born with additional legs or wings, but those extra limbs are mostly useless because they are not fully functional.
Hypothetically, if there would be an animal that gains fully functional extra limbs through polymelia, that animal could have an advantage in the evolutionary competition, thus it would survive and give the genes responsible for polymelia to it's descendants.
Such an animal with two extra arms could have been the ancestor of dragons. One pair of arms would have evolved into wings later.
Wings made of bony rods
Maybe, the wings of dragons are no real wings. Over the history of the earth, there where many kinds of reptiles who had elongated ribs or bony rods protruding from their chest that supported gliding membranes. This didn't give them the ability to fly, but to glide from tree to tree. Most of those animals where not directly related to each other, and with Draco, there is still one genus of lizard that glides this way.
Maybe it's possible that a set of muscles evolves that allows the animal to flap those pseudo-wings, thus allowing it to actively fly.
Many legends about dragons tell us that the belly is the most vulnerable part of a dragon's body. That makes sense, since the ribs have evolved into wings, so dragons lack a chest that protects vital organs from harm.
Main article: Wyvern
Wyvern are dragons with only one pair of legs. It's not unknown in evolution for animals to evolve wings and gaining the bodyshape of a wyvern. In fact, every flying vertebrate known to science (birds, bats and pterosaurs) have one pair of wings and one pair of feet, essentially making them wyverns. There was one species of birdlike, gliding dinosaur called Yi qi, that really reminds of a wyvern because of it's bat-like wings. Maybe, every dragon is a wyvern and there are no six-limbed dragons.
This theory is most likely, because having four forelimbs wouldn't leave much place for the powerful chest muscles required for flight, so maybe four-legged, winged dragons couldn't fly at all even if they existed.
Some scientists suggested that dragons are real hexapods (naturally six-limbed animals) that descended from six-finned fishes. This would make them only distant relatives to tetra pods.
The biologist M. May suggests that dragons, along with other six-limbed beasts of legend such as griffons, pegasi, centaurs and angels, form their own sister-clade of tetrapoda, while wyvern and the Cockatrice are tetra pods.
This theory is highly unlikely, because there is no known species of hexapod vertebrate and also no known six-limbed Sarcopterygii. The reason for this is that the evolution of paired fins took place very early in the history of fishes, and only four-finned fishes have survived since then.
On a few instances, dragons have been portrayed as arthropods, mostly as insects. This is also highly unlikely, because the atmosphere doesn't contain enough oxygen to support arthropods this big. Also, most mythological dragons are definitely vertebrates, mostly reptiles. The only insectoid dragon from mythology is the Pyralis, while most fantasy-versions are based on the word dragonfly.
More information on the insect-theory is provided by Sviatoslav Loginovs in his article Sviatoslav Loginovs On Classification of European Dragons.
It should also be noted that, while insects have six limbs, like western dragons, their wings are not limbs, so they still have six legs, while dragons typically have four legs.
In some works of fiction, dragons are portrayed as extraterrestrials. In this cases, on their planets it's totally normal for a vertebrate-like being to have six limbs. One example for this is the Draco berengarius from Star Trek.
Size and weight
Most fantasy-dragons are giant beasts, which would make them unable to fly by their weight alone. The biggest known animal to fly is the pterosaur Quetzalcoatlus which may had a wingspan of 12 m, but since the wings make up the majority of the body, it was very lightweight (100-200kg). Dragons, however, are much bulkier, which would also make them heavier.
The following adaptions could possibly make them light enough to fly regardless of their massive size.
Like every other flying animals, dragons for sure have hollow bones, because bones make up most of the body's weight. Having hollow bones allows birds, bats and pterosaurs to fly and is a necessity for flying vertebrates.
The most common theory in dracology is the existence of a flight bladder. The flight bladder is an organ that only dragons have, similar to a second pair of lungs. In this organ, the dragon collects gases from digestion. Most theories also suggest that the gases in the flight bladder allow the dragon to breathe fire. The first to theorize about a flight bladder is the documentary Dragon's World - a fantasy made real. 
There is a similar organ in fishes, the swim bladder, which gives them buoyancy while swimming. The lungs of vertebrates evolved from the swim bladder.
The first to make a similar theory was Peter Dickinson in his book The Flight of dragons. He suggested that dragons don't have ribs, because those evolved into wings, yet the body keeps its shape because of the pressure of gas inside the dragon. This is unlikely because it would make the dragon a living gas-balloon.
According to Joschua Knüppe, the lungs of dragons (Draconiformes) have increased in size in comparison to their relatives. This leaves less space for their entrails, which is why they have to dissolve their food with Dracotoxine.
Also, according to some theories, the bodies of pterosaurs where filled with systems of air-sacs to make them more lightweight. There's a fossil whose spongy structure undermines this idea. Dragons could theoretically use a similar system.
Also, Henry Gee states that all kinds of archosaurs, for example crocodiles or sauropodes, had a similar system of air-sacs throughout their bodies to help them cool their bodies. In the case of sauropods and birds, this system could also make the body lighter, in case of the birds to help them with flying, in case of the sauropods to assist their giant sizes. Gee suggests that dragons could have such a system because it would be more effective than a flight bladder. Also, he assumes that the scales of dragons help them to reduce air drag while flying. This system of air sacs could be a basal trait of archosaurs, which maybe includes dragons.
Another theory suggests, that dragons simply aren't as that big. Most medieval illustrations depict them as small, maybe the giant dragons of fantasy are just exaggerated.
This idea matches with the theory that dragon-sightings are inspired by big birds. These birds also wouldn't be as big as typical fantasy-dragons.
According to Leonardo Da Napilut (who may or may not exist), dragons create their own thermal column, using their fire-breath.
A similar theory comes from Hugo de Folieto, who believed that the venom of the dragon disturbs the air, so that the dragon is lifted. During the middle ages, these theories were somewhat widespread, so it's also stated by Bartholomaeus Anglicus.
To allow the dragon to fly, the anatomy of the tail has to be adapted to flight. Most dragons are depicted as snake-like animals with long tails, some dragons (like the Peluda or the dragons of India) are able to constrict their prey with their tails.
In contrast, the flying animals of today, birds and bats, have rather short tails. In case of the birds, the tail feathers form some kind of fan to steer while flying, while in bats, the hind legs, connected by loose skin, take on that task. The extinct pterodactyloid pterosaurs also use their hind legs, but in a different manner than bats.
The reason for this is that a long tail creates drag, which makes the long tails of many dragons highly unlikely.
Yet, there once existed flying animals with long tails. These were the rhamphorhynchoid pterosaurs (all non-pterodactyloid pterosaurs) and several species of Paraves, including some extinct birds.
All those animals had in common, that their tail was made stiff by so called chevrons, small bone rods protruding from the vertebrae. Because of those, the tail could only be moved lateral, but never dorsoventral.
Naturally, animals with long tails also need to steer in the air. For this, rhamphorhynchoids have a tail vane on the tip of their tails, while early birds had feathers just as their modern cousins. In modern fantasy, most dragons also have a tail vane or some kind of fin made from skin that could be used as steering vane.
However, the tail has to be stiff to grant stability while flying, so it's highly unlikely that they are able to constrict prey with their tails like some snakes do. In general, the tail of a flying animal wouldn't be a very good weapon. Supporters of the Dragonbird-hypothesis suggest that the idea of a snake-like tail is based on sightings of the long tail-feathers of pheasants and similar birds.
It should also be noted, that all long-tailed flying animals are rather small, because for big animals, the created drag would make flying very difficult if not impossible.
Another comment to the serpentine body comes from biologist Peter John Hogarth. He thinks, that dragons have evolved from serpentine ancestors over the last 5000 years. In this process, the body has become more compact and less serpentine. This is mirrored by dragons in art, which where long and serpentine in ancient Sumerian and Greek art and later became shorter and gained wings.
While Hogarth argues from a mythological point of view, this theory could also be true for the biological evolution of dragons. In this case, the serpentine tail of early dragons evolved into a stiff tail to enable them to fly. Some flightless, serpentine species could have survived long enough to inspire art of early cultures.
Maybe, all dragons are flightless. The idea of flying dragons could stem from wing-like sails or feathers on the dragons backs that have been mistaken for wings, or useless vestigial wings inherited from smaller, flight-capable ancestors.
For example, Oriental Dragons are seldom depicted with wings, but they have structures above their legs that look like feathers and could have been mistaken for wings. According to most myths, these wingless dragons are capable of flight.
It has also been stated that dragons have scales that cannot be pierced by swords. Heavy armored hide would be hindering to flight and makes it unlikely that dragons could fly.
- Wikipedia: Polymelia
- Drachen-Kompendium: Drachenbauch (German)
- Dragons Nest: Robert M. May - the ecology of dragons
- Biology letters: Unusual anal fin in a Devonian jawless vertebrate reveals complex origins of paired appendages
- Sviatoslav Loginov: On Classification of European Dragons
- Dragon's World (2004) in der IMDB
- Peter Dickinson (1981), The Flight of the Dragon, HarperCollins, ISBN 978-0060110741
- Hyrotrioskjan: Dracotoxine - the real dragon fire
- Scienceblogs: Pterosaurs breathed in bird-like fashion and had inflatable air sacs in their wings
- Discover-Magazine: How to explain your Dragon
- Heise.de: Luftige Dinosaurier (German)
- Boogleech: Creepy Classic Dragons
- Amaranon Magazin, Ausgabe #3, Artikel: "Drachen und ihre Flugfähigkeit - Eine wissenschaftliche Annäherung" (German)
- Medieval Bestiary: Dragon
- TREY the Explainer: Cryptid Profile - The Ropen or "The Living Pterosaur"
- Pterosaur.net Blog: Dragon Tails: What Pterosaurs Teach Us about Velociraptor by Dave Hone
- Frontiers of Zoology: Gwiber Wyvern by Tim Morris
- Peter J. Hogarth: Ecological aspects of dragons