There are scientific reasons (along with a dose of common sense) as to why a man cannot just attach a set of wings and fly like a bird. But in mythology, the Bible and in the minds of some artists, angels (and mortals) do have wings and can fly. I’ll discuss why.

If you wonder why man cannot fly, consider in physics the biology of birds versus the human body. Our body is not streamlined and our center of gravity would be poor if we were to attach wings. Also, birds have a keel shaped sternum (breastbone) which is lacking in humans, so that is why we are meant to stay on the ground.

If you are not yet convinced, consider how the feathers of birds are protection from harsh weather, the sun, and parasites. And besides, the energy that it takes to take flight would consume more energy than a human could possibly muster.

As early as the creation of Byzantine art in the 13th century, angels were depicted. Most significantly, they influenced the Italian artist Giotto from the Middle Ages. His angels from the fresco "The Lamentation" (1305-1306) are especially memorable as they wear gilded halos and their arms and bodies are small in proportion to the size of their heads. The beauty of these angels lies in their personalities and emotion: sadness at the lamentation (mourning at the death of Christ) in this case.
"The Lamentation" can be seen at the Arena Chapel in Padua, Italy.

The only painting of Greek mythology by the Italian artist Raphael was "Triumph of Galatea" (1512) for the Villa Farnesina in Rome, Italy. It depicts sea creatures and angels shooting or holding arrows above Galatea, who represents ideal beauty. Some critics believe Raphael’s use of arrows has the hidden meaning of paint brushes, thus including his profession into the work other than a self portrait.

One of my favorite stories from Greek mythology is from the Ovid, the story of Icarus. Decided in the late 20th century to be from the "circle" of Peter Bruegel the Elder is "Landscape for the Fall of Icarus" (1560). Icarus succeeded in flying with wings made by his father until he flew too close to the sun. The feathers on his wings were secured with wax; they melted and he fell into the sea and drowned. I think that should be the deciding factor to dissuade man from taking flight anytime soon.
This outstanding painting, now considered an early copy of Bruegel's original, can be seen at the Museum van Buuren, Brussels, Belgium.

You can own a giclee print of "Landscape for the Fall of Icarus" from the circle of Peter Bruegel the Elder.