Can I set my pose position like the new rest position? I would like to move my armature from pose mode to another pose (a random one) and set the position I create like the new rest position. Is there a way in order to achieve this goal?

My armature is deforming a mesh so I want to deform the mesh too.

This is my mesh and its relative REST POSITION:



I want to change and set a new REST POSITION and I follow your advice; 
I change my armature in pose mode and then click on Apply pose as Rest Pose:



The result is this one:



The armature resets the rest pose like I wanted, while the mesh - I can't understand why - comes back to the original position (the one set by the old rest pose).

Why doesn't it work?
Apply the armature modifier - armature in rest pose doesn't deform anything, so we need also a mesh to match the new rest pose.
Pose > Apply Pose as Rest Pose. This will set current pose as the rest pose. It sets only bones poses - constraints or custom properties in complicated or badly designed rigs might make the rig act weird with bones in different rest state!
Add the armature modifier back.