Do photons have acceleration? Photons travel at the fastest speed in our universe, the speed of light.
Do photons have acceleration?
 A: By definition, acceleration is a change in velocity. Velocity is a vector, so it has a magnitude (also known as speed) and a direction. For light, the magnitude does not change, but the direction might.
For example, consider a light ray bouncing off a mirror. The incoming light is going in one direction, the outgoing light is going in another direction, so presumably there is some acceleration involved. However, actually calculating the acceleration gets a little tricky because $\vec a_\text{avg} = \frac{\Delta \vec v}{\Delta t}$, and it's hard to precisely define $\Delta t$, the time that it takes the light to bounce off the mirror. Once you look closely enough, the action of reflection starts to seem more like absorption and re-emission.
You might also ask the same question about light traveling in empty space, free of anything to reflect off of. Here it gets a little tricky though, because light traveling in empty space might or might not accelerate depending on how you define "constant velocity."
In Newtonian physics, we tend to define "constant velocity" relative to the Earth, or the sun, or some other convenient object. In this view, the direction of a light ray (and thus its velocity) will change as it passes by some massive object and is attracted by gravity. So when you look at it that way, light does accelerate in empty space.
But in general relativity, "constant velocity" is defined in terms of parallel transport - not relative to a convenient object, but relative to the way the object's path "naturally" extends itself through spacetime. In this sense, the "natural" extension of a path is a geodesic, precisely the kind of trajectory that an object (or a light ray) follows under the influence of only gravity. So in the general relativistic sense, light rays do not accelerate in the absence of materials.
A: Gravitational lensing has been observed which means that the photons bend. An acceleration can be defined in its change of direction, angular acceleration in radians/second^2, so the answer is positive, yes, light can be accelerated, but its speed  will still be c locally.
A: In the vacuum, the light follow the minimun distance of the space-time even if the is the massive object (like gravitational lense).Light alway goes right away according to the space-time metric.  Photon doesn't  loss energy => it not accelrate/decelerate
A: The speed of light is a constant, but it can accelerate by changing direction. For instance, light accelerates under the influence of gravity like every other falling object. 
Reflection is not a very meaningful form of acceleration since the $\Delta t \approx 0$, but it certainly a change in velocity. 
Light can also gain and loose energy like other accelerating objects. It does this by changing wavelength (red/blue shifting).
A: If the redshift factor differs from the change in the rate of change of affine parameter under parallel transport, yes.
A: I think you mean to ask whether a photon can accelerate. The answer is yes. For instance when travelling from one medium to another light changes speeds. In fact, the actual photon is abosrbed and a new photon is reradiated during this process. However, you could call this effectively light acceleration
