Whenever I move an object by the transform manipulator in the 3d view, it disappears when I hold it (like normal) but when I let go, the arrows haven't followed the object. It just sits in the center of the scene no matter what.
I assume the Pivot Point for rotation and scaling is set to 3D Cursor. In this mode the 3D manipulator, i.e. the movement arrows, is also placed at the 3D Cursor.
Setting the pivot point to something like Active Element will also move the 3D manipulator to the origin of the active object.
Alternatively, depending on your intentions, you can move the 3D cursor, for example to the active object by pressing Shift+S and selecting Snap Cursor to Active.