Has anyone made LEGO fly?
Non powered/motorized flight has been achieved

Bamboo copter

Anna Vuurzoon made a Flying machine using only Lego parts which uses a ripcord to propel a three blade propeller on a stick (see http://www.mocpages.com/moc.php/288875). I was able to duplicate her results, so can validate it does work. It is really a modification of a bamboo copter. The 2015 Ninjago Lego Airjitzu series 
( Lego #70739, 70740, 70741, 70742, 70743, 70744) are bamboo copters and work on the same basic concept as Anna Vuurzoon's flying machine.

Glider design

Daron Williams (via Drat Glider) and Milan "grohl" Reindl from Czech Republic have both reported making working/flying Lego gliders using mainly Technic cross-axles components.  Both gliders travelled between  20 to 30 feet with the Drat Glider with the bigger wingspan going a bit longer.  The wing cross section is more or less flat (one stud tall airfoil) and acting more like an air sail (paraglider) than an airfoil. Williams uses Lego Plastic sheets from the Mars Exploration Rover Set to cover the wing (wing cowling). Plastic sheets is the lightest wing covering that can be used but Lego does not make very large ones. Reindl uses plastic from garbage bag and tape to cover the wing on his glider - this allows for a bigger but heavier cowling. I have made my own Lego glider using the plastic from Lego store bags as my wing canvas using similar construction techniques as Williams and Reindl . My preliminary indoor flying test validate the results by Williams and Reindl.  Milan Reindl is now a Lego Group junior designer (you can see his profile on Lego Technic Designer Profile).  

http://www.mocpages.com/moc.php/1769

http://www.youtube.com/watch?v=3CQcCoZ9rcY

The Challenge: 

Lego does not make a quality aerial propeller or water/hydrodaulic screw-prop with a fuel efficient power stroke. The aerial propeller design for an all-electric aircraft have already be optimized by the Radio control aircraft community and can be fitted to a Technic axle but modern aerial prop design need to run from 4000 to 10000 rpm and use poly lithuim batery packs and motors that run at higher current and voltage than the Lego power function components. 

The Vuurzoon design takes a lot of torque/power to fly and uses the power that it is given inefficiently. The only thing going for the Vuurzoon prop design is that it has the most efficient rotor angle to create lift. The Vuurzoon design uses hand power instead of electrical power and thus avoids the low power limitation of Lego electrical motors and battery pack systems.  Lego motors without gearing don't go above 2000 rpm at their set voltage/current level and have a high weight to power ratio which is bad for flying. Going over an 8:1 gearing causes a Lego motor to lose so much power/torque that any propellers slow down due to wind resistance (even the most energy efficient  light weight two blade electrical propellers has this problem).  Lego motors can go a bit faster if pumped with higher current and higher voltage from a non standard non-lego power source (e.g. poly-Lithium +15V power pack) but many believe this will shorten the motor lifespan.  Lego Power Function electrical packs are also too heavy for flight  the lightest battery pack is a Li-ion rechargable pack which weights about 75 grams. The heaviest Lego battery pack is the AA battery pack with alkline batteries which weights 195 grams&nbsp;— which is enough to sink a two catamaran hull or a large single hulled lego boat.
Standard Lego bricks are too heavy for an heavier-than-air airframe. The lightest Lego parts that can be used as an air frame are the technic axles followed by the half width technic beams— both are only semi-ridge so some bracing is need if air frame rigidity is required. In order to get a battery pack light enough and with sufficient voltage levels to fly as part of the aircraft&nbsp;— Lego needs to have to Lithium polymer battery pack kicking out between 18vdc to 30vdc&nbsp;— at those voltage levels the mini hi RPM motors used on Quadcopters and helicopter toys can start to be posssible.  

Analysis of other attempts

Lego Quadcopter

There are several efforts to use Lego to make the frame of a quadcopter and uses a non lego high speed lightweight motors, hi powered batteries, and electric propeller. The high speed  motors usually run at 3400 to 15000 rpm @ 13 to 30 vdc and are very light weight. In addition it is using a non lego Poly-Lithium power pack some output as high as 30vdc even with heavy current drains. Some are using a Arudino microprocessor like device for remote communication. 

http://www.youtube.com/watch?v=DkrTJh0H1fs

http://robotsquare.com/2012/03/02/flying-lego/

http://tinkernology.blogspot.com/2010/08/lego-nxt-quadcopter-no-really.html

http://www.suasnews.com/2012/12/20276/drone-made-of-lego-takes-flight/

Lego NXT Blimp-Airship

A lighter than air Lego Airship which was made by The Lego Group for a European Lego show/convention&nbsp;— uses Lego education propellers (from the renewable energy kit) and standard Lego motors and power supply. Prototype (non-commerical, custom made) helium balloons are used to create a neutral buoyancy so that the propellers can create lift and push the airship forward.  The Lego airship (called the C5) was demonstrated in an auditorium/stadium with a very high ceiling (there is no wind of any kind that the Airship would have to overcome).    

http://www.youtube.com/watch?v=GefwcBUioAE