Does gravity sometimes get transmitted faster than the speed of light? Consider Earth moving around the Sun. Is the force of gravity exerted by Earth onto the Sun directed towards the point where Earth is "right now", or towards the point where Earth was 8 minutes ago (to account for the speed of light)?
If it's the former, how does the Sun "know" the current orbital position of Earth? Wouldn't this information have to travel at the speed of light first?
If it's the latter, it would force a significant slowdown of Earth's orbital motion, because the force of gravity would no longer be directed perpendicular to Earth's motion, but would lag behind. Obviously, this isn't happening.
So it appears that the force of gravity is indeed directed towards the current orbital position of Earth, without accounting for the delay caused by the speed of light. How is this possible? Isn't this a violation of the principle that no information can travel above the speed of light?
 A: It seems that your intuition is tricking you. If I understood you correctly, you're thinking of the Sun's gravitational pull as directed at the Earth, like it's pointing at the specific "target spot" in space, and when gravity reaches that spot, it pulls there.
But it's better to think of gravity as pulling from all directions at once: so the Sun is simultaneously pulling both the spot where the Earth is now, and the spot where the Earth will be eight minutes from now, and every spot in between, and every other spot in our orbit, and so on.
Think of it the same way you think about sunlight: the sun is shining in all directions, so it never needs to know where we are going to be - some light just goes out in a straight line in all directions, and so we always miss the light that was headed towards us when it left the sun, and hit the light that left the sun pointing at where we would end up eight minutes later.
In this case, you can think of them the same way: The sun isn't sending those things towards us specifically, it's just sending them out everywhere at once.
A: I was going to write an incorrect answer at first, but having researched a bit
what Yukterez wrote, I think I may be able to offer some intuition.
First, let's look at the solar system in the centre-of-mass frame, where the
Sun is essentially stationary in the middle. Here, the force on the Earth is
directed towards the Sun's position 8 minutes ago, which conveniently is the
same as its current position. So this doesn't affect the Earth's position in
the way you say.
If the Sun and Earth were moving at a constant velocity to each other, they
would, like Yukterez wrote, still be attracted to the instantaneous position of
the other body. This is not because of a faster-than-light influence, but
because gravity is more than a simple attractive force between objects
(analogous to the electric field in electromagnetism). For moving sources,
there are other components of the gravitational field (roughly analogous to the
magnetic field). You can view this as the force travelling at the speed of
light, but being directed towards the position of the source "predicted" based
on its past linear motion.
However, if the source is accelerating, like the Earth around the Sun, the
effects are less obvious. Then, you really can't view it only as an attractive
force towards any position (I think, correct me if I'm wrong), but you get
more complex things like gravitational waves. But still, nothing ever travels faster than
light.

As I am more familiar with electromagnetism, let's take an example from there.
If two oppositely charged bodies ($A$ and $B$) are moving with constant
velocity towards each other, we can view the system in the rest frame of $A$.
Here, it emits only a static electric field, given by the Coulomb formula, and
$B$ therefore feels an attraction directly towards it. In the rest frame of $B$,
this means that the attraction is towards the instantaneous position. But how
can this be since the electric field is propagating at a finite speed? The
answer is that since now $A$ is moving, it also emits a magnetic field. While
this doesn't affect $B$ directly (since $B$ is stationary here), it does affect
the electric field and makes it point in a different direction (towards the
instantaneous position).
A: Gravity is the bending of the space and the sun's mass causes bending of the space around it and in a circular path around the sun it is uniform. The earth travels in elliptical orbit so the force of gravity is more at some places i. e.  the bending of space is more at a place closer to the sun and less at a place far from it. The force of gravity is not exerted by the sun on the earth but the bending caused keeps the earth near the sun and keeps it revolving and the velocity ensures that the earth does not just fly away in the direction of the tangent drawn to its orbit. So there is nothing like the sun "knows" where to exert the gravity but as I have said that it is uniform in a circular orbit so the bending is the force of gravity acting on it. The angle between the earth's motion and gravity(acting as the centripetal force here) will be 90°.The earth's motion will be in the direction of the tangent drawn at the orbit. As you might know that the centripetal force acts towards the center of the circular path and the angle between a tangent and the line passing through the center is always 90° so that law is never violated. 
A: No, gravitational influences never travel faster than the speed of light. However, a naive incorporation of a speed-of-gravity delay would actually lead to the Earth's orbital motion speeding up, not slowing down. (Think about the geometry carefully.) I explained here why that doesn't actually happen in general relativity.
A: 
Cuckoo asked: So it appears that the force of gravity is indeed
  directed towards the current orbital position of Earth, without
  accounting for the delay caused by the speed of light. How is this
  possible?

If the motion is straight or circular the aberration cancels out, see Steve Carlip: Aberration and the Speed of Gravity:

Steven Carlip wrote: The observed absence of gravitational
  aberration requires that "Newtonian'' gravity propagates at a speed
  ς>2×10¹⁰c. By evaluating the gravitational effect of an accelerating
  mass, I show that aberration in general relativity is almost exactly
  canceled by velocity-dependent interactions, permitting ς=c. This
  cancellation is dictated by conservation laws and the quadrupole
  nature of gravitational radiation.

or to quote the Wikipedia article on the subject:

Wikipedia wrote: Two gravitoelectrically interacting particle
  ensembles, e.g., two planets or stars moving at constant velocity with
  respect to each other, each feel a force toward the instantaneous
  position of the other body without a speed-of-light delay because
  Lorentz invariance demands that what a moving body in a static field
  sees and what a moving body that emits that field sees be symmetrical.
  In other words, since the gravitoelectric field is, by definition, 
  static and continuous, it does not propagate.

A: Let's temporarily pretend that we can speak of gravitation as being released in pulses, which is a weird way to speak, but I think is part of your mental model.  Let's also pick a reference frame : suppose the Sun is stationary in our laboratory.
The Earth right now is not reacting to the right now pulse of the Sun's gravity.  The Earth right now is intercepting the pulse of gravity released $8$ minutes ago, which happens to point directly back to the Sun (because it is stationary).
For a careful, mathematical treatment of this, see The Feynman Lectures in Physics, vol. II, section 26-2, where an electric field in laboratory coordinates is found to have magnetic components in the moving frame which (to first order) cancel the aberration (angular deflection) caused by retarding waves by their travel times.  The same thing happens in gravitation: the off-diagonal elements of the tensor pick up the terms necessary to cancel retarded aberration (to first order).
A: The earth moves in the static, unchanging gravitational field of the sun. Its potential has the form $V(r) = -G_N M/r$. This field acts on all objects exactly the same way, and the force is equal to the gradient of this potential times mass of the object: $F=mg$ and $ g= -{\partial V \over \partial r}$. The gravity, g, has magnitude and a direction and it points toward the sun!
 Why not accept this simple, correct, standard answer?
A: Think of the gravitational field as curved space. Like a bump in space time. This bump will stay constant over time more or less. The planet then just looks at the bump and moves accordingly. However if the sun were to explode it might cause a small ripple in spacetime that would reach the earth in 8 minutes
A: In Newton's theory of gravity, the gravitational force is defined as being proportional to the product of the two attracting masses and inversely proportional to the square of the distance between the masses. The force is instantaneous. As Newton said himself, this is philosophically unsatisfying - for example, at any instant, how does Earth know how much mass the Sun has, how does it know how far away from the Sun it is, to know what gravitational force is being applied? But Newton's theory works and is very accurate. 
It is in the theory of General Relativity that signals can't travel faster than the speed of light. This is carried over from the Special Theory of Relativity along with the difficult to visualise idea of 4 dimensional space-time. In outline, General Relativity defines gravity in a different way, there is no force of gravity as such. Bodies like Earth follow a 'straight' path (or geodesic) through space-time. Far from gravitational sources space-time is flat, at a gravitational source space-time is stretched or squeezed, in between the space-time must smoothly transition from squeezed/stretched to flat. Bodies like the Earth follow the straightest path through space-time at each local point. [Note, General Relativity does not say that mass (or strictly speaking the total mass/energy) causes space-time to squeeze/stretch but defines the correlational between the two). 
To good approximation the Sun is symmetrical, and doesn't change over short time spans, and the space-time around it is not squeezed or stretched much (it is a relatively weak gravitational source). This means that the shape of space-time near the Sun is static and there is no issue with changes to the space-time having to be propagated from the Sun to the location of the Earth. In General Relativity the Earth follows the 'straight' line through space-time locally and so avoids the philosophical problems in Newton's theory.
I think this is an interesting question, it draws out that different theories define physical properties in different ways (gravity in this case), that although Newton's theory of gravity seems simple on the surface it can be, at least philosophically, hard to understand.
A: The sun exerts gravitational waves in all directions, not just at other bodies of mass like the Earth. When the Earth arrives at some point in space, it is attracted to the Sun by gravitational waves of the Sun that are already there. 
For example, say the Earth is at point A and will arrive at point B in 1 minute. When it gets at point B, it will be pulled by gravitational waves that have already been traveling for 7 minutes from the Sun to arrive at point B.
