How to move an object along given normal after applying rotation I know that one can move rotated object along one of the normals' axis by G + Z + Z (or whichever desired axis). I also know that after applying rotation, a face or edge can be moved in similar manner in edit mode. But if you selcet all of that object elements, it can only be moved along its general axis (which is different fro the default world axis, because the rotation was applied and the values of XYZ were reset to 0). 

My question is, how to move an object along one of the normals of that object, in object mode, after ctrl + A -> apply rotation. Let's say I have an object, I cut out and duplicate a piece of it, I then apply rotation and want to move it along one of the titled / rotated normals of that object. For the life of me I cannot figure this out. 

Any helps is greatly appreciated. I am not uploading a blender file, because I do not think it is necessary in this case, but I will if it is needed.
You can create your own axis selecting at least 3 vertices, then ctrl alt space, and in the bottom horizontal menu bar you'll see this new axis appears as one of the Transformation Orientation axis. You can manage these new axis in the 3D view properties panel (enabled with N) > Transform Orientations.