Speed of Dark experiment, does it *really* travel faster than light? I have seen many folks doing this Moon shadow experiment concluding that shadow from point A on the Moon can travel to point B on the Moon faster than light.
What I fail to understand here (and I am sure I am wrong), is nothing can travel faster than light. I also read about Relativity where Einstein stated that every event happening for you depends on how fast the light gets from where the event is happening to you.
This is my particular problem with this Moon shadow experiment. You cannot make a shadow travel faster than light, because in order for the "disappearance" of light to get to the Moon, it has to travel at light speed.
Here is one way to understand my point. Consider we shine a flashlight on Moon, like the guy in the linked video did, we move the finger across the face of the flashlight, before we moved the finger it was casting the shadow at Point A on Moon, after we finish moving our finger, the shadow is at Point B on the Moon.
When the finger reaches the end of it's moving length, the light waves that have left the flashlight before the finger reached there will have to hit Point B and it will take them a second and a half to get there, and another second and half for us to see that shadow on point B, which makes it 3 seconds + finger moving time for us to actually see the shadow at point B. That means shadow traveled (say) across the diameter of the Moon in 3 seconds. 
So. about 3000 Km in ~3 seconds make it 1000 Km per seconds which is very tiny compared to the speed of light. How can these experiments conclude that Shadow could be made to travel faster than light? What is the obvious clue that I am missing here?
 A: I think you've said a lot of things that are correct, you've just come to the wrong conclusion.
You're right that that it will take us three seconds to SEE the shadow moving across the moon, because that's how long it takes the light to get there and back. But what we'll see is not the shadow slowly moving across the moon over a period of three seconds. What we see is: nothing happens for three seconds (the shadow stays at point A); then, the shadow races across the moon to point B, possibly exceeding the speed of light.
The reason this can happen is simply that shadows are not made out of particles, so there's no reason to think they can't go faster than lightspeed. And as @brucesmitherson pointed out, there's actually no information being transmitted faster than light, since the shadow still takes three seconds to start moving.
A: It is known that moving a lantern will result in a spot or projected light that travels faster than light is the screen is far enough. However, this effect and similar ones do not refer to the motion of actual objects at faster than light speeds, nor this process allows
 transfer of information faster than light.
A: Imaginary things can "travel" faster than light
A shadow or a light spot can seem to travel faster than light, because it's not a particular physical thing, but a series of separate things, separate physical particles emitted at different time and at different locations.
Imagine that you have launched a lot of tiny bots into space with a very accurate clock and a single LED, spaced out in a straight line with a 1 km distance between each of them.
If you program them to blink their LED at particular times – say, the first one blinks at midnight, the second one at midnight + 1 second, the third one at midnight + 2 seconds, then you'd see a spot of light moving at 1 km/s across this line. If you program them to always be on except for a particular moment arranged in the same manner, then you'd see a "shadow" moving at 1 km/s.  
If you'd do the same, but set the intervals when your bots light up to 1 millisecond instead, you'd see that the signal is "moving" at 1 000 km/s. If you would have them light up at 1 microsecond difference between the neighboring bots, almost at the same time, then you'd see that the signal is "moving" at 1 000 000 km/s, much larger than the speed of light – but note that there is nothing that's actually moving there, the bots are stationary.
The same applies for true shadows – they're reflected off of something that's not moving (as much), the reflected photons for each moment are different reflected photons, and the fact that a moment ago there was a reflection much further – the "reflection has moved" at above speed of light, describes only the distance and time between two separate events, not an entity that has moved anywhere.
