I try to move meshes but the just won't translate. I don't know if I hit any key. The translation values just vary between 0 and - 0 on any axis
The most common issues may be either you have some transform channels locked, see your options bar (N) :
Or you have a Limit constraint on your object, in which case you just have to delete it :
Or your transform properties are driven, in which case you just have to right-click → Delete Drivers:
If it neither those issues, please provide more info in your topic, as well as your .blend file.