Other than those with obvious mechanical assistance (either wings or rockets), flying characters possess the ability to propel themselves through no effort of their own. This might be due to some inherent telekinetic ability, or perhaps it might be due to the ability to manipulate their own personal gravity. Winged flyers don't have it much better; they need some way to offset the mass and awkward shape of the human body, whether it be super-powerful wing muscles (and more importantly very large wing surface area, about that of a hang glider would do), a bird-like hollow bone structure (which would only save a few pounds for something with the body mass of a human - some engineering doesn't scale up well, as the ostrich can attest), or perhaps even a method of non-powered flight or levitation that is merely augmented by the wings (or some combination of the three).
Also, for those that go high into the atmosphere, they also have a reduced requirement for oxygen. As there is less atmospheric pressure higher in the atmosphere (one's lungs operate by manipulating internal air pressure to move air into and out of the lungs), flying characters would either need to stay closer to the ground, be able to generate a higher partial pressure of oxygen from their surroundings (via air-related powers or molecular control) or possess less need to breathe (and thus could stay high up for long periods of time). Presumably, this would also lead them to hold their breath longer at sea level, but even if they can breathe if they have winged-based flight a lower atmosphere pressure would mean less lift so they would need very big wings in order to reach high altitudes.
Thirdly, there is an issue of temperature. The adiabatic lapse rate is (in layman's terms) the rate at which the temperature decreases as you go higher above the earth's surface. A flier must have some way of preserving body heat at altitude or they would freeze in short order.
Some of the same problems associated with speed also apply to fast fliers. Flight almost invariably involves a combination of Super Strength and the ability to ignore aerodynamics, as fliers often lift off while carrying someone or something. One also has to wonder how they don't destroy half the atmosphere flying that fast. The shockwave and heat would be devastating - Tunguska style.
In addition, it has been pointed out that many fliers must also have some regenerative powers. Otherwise, the constant effects of the wind and airborne dust particles on exposed skin would cause premature aging. Storm for example should have skin resembling leather and more wrinkles than women decades older due to constant exposure to fast moving wind, rain, and snow. On a related note, there must be something special about fliers' hair, otherwise most of them would be sporting horribly tangled locks after each flight. Even more impressive is the way it never seems to be blown into their eyes.