Is it really impossible for Bumblebees to fly? According to some source or other (I forget which now) it is theoretically impossible for bumblebees to fly by virtue of their size/bulk/aerodynamic properties. Is this old adage apocraphyal or true? And if its true, how come they do fly?
 A: I would just like to add something here.
These answers are great but I might have another answer.
At first it was indeed a mystery how these insects were capable of flying, but thanks to high speed recordings they found something which investigators didn't consider. The wing motion has a sort of double lift feature.
By twisting her wings over at the end of each down stroke, the upward momentum is never lost.
Basically, this means that even when the wings are going upwards, the wings provide upward lift.
(this also explains why, because bumblebees are so massive, they need a lot of muscle to get at the needed (Ca.) 300 flaps a minute)
I hope this was a complete answer!
Cheers.
A: This story may have originated with August Magnan and André Sainte-Laguë. In the forward to his book Le vol des insectes, August Magnan wrote 

Tout d'abord poussé par ce qui se fait en
  aviation, j'ai appliqué aux
      insectes les lois de la résistance de l'air, et je suis arrivé avec M.
  Sainte-Laguë a cette conclusion que
  leur vol est impossible.
First, prompted by what is done in
  aviation, I
  applied the laws of air resistance to
  insects, and with M. Sainte-Laguë came to the conclusion
  that their flight is impossible.

(my translation ... please correct if it's wrong).
He is talking about insects, and not specifically bumblebees.
A: The key is to correctly take scale into account. At insects scale, the Reynolds number is low, meaning that flow around an insect wing is mostly laminar, not turbulent. Moreover, kinematic viscosity of air is higher than water : at their scale, insects do not fly like birds, they litteraly swim in air.
More information:


*

*http://quest.nasa.gov/people/bios/aero/fjournals/wellman/bumblebee.html

*https://en.wikipedia.org/wiki/Insect_wing#Aerodynamics
A: No it's an urban myth. It's impossible for them to fly using a very simple and inappropriate model of wing behaviour - possibly closer to say that bumble bees can't glide like albatrosses
A: Insect flight is different than bird flight. With insects, the rapidly moving wings, which do a figure 8 sort of motion, generates a vortex tube over the wings. This vortex by Bernoulli principle has less pressure, which permits the larger air pressure underneath to lift the animal up.  If one is trying to understand insect flight according to the mechanics of bird or aircraft flight you are then pounding a square peg into a round hole, which makes you conclude bees can’t fly.
A: Newtonian mechanics based on the mass flow rate resolves the controversy how a bumblebee can fly with relatively small wings compared to its big body and mass.

On each wing cycle, the bee’s small wings pass through a small mass of air (small ‘m’). A high wing beat frequency means that the mass of air pushed down each second is modest (modest ‘m/dt’) relative to other insects.
Then the critical point that differentiates bees is that they aggressively accelerate this air downwards (high ‘dv’) with their large muscles to generate a lot of lift. This explanation is similar to how a heavy fighter jet generates significant lift despite its short wingspan; by flying fast (modest ‘m/dt’) and aggressively pushing the air downwards (high ‘dv’).

