Walls and ceilings
Dust off all walls and ceilings with a dry cloth. For ceilings, it's best to use a broom around which you have wrapped a soft cloth or gauze. With this method you'll be sure to grab even the smallest spiderweb.
Wipe down emulsion coatings with a wet cloth - only damaged areas or those which have been painted a different colour should be repainted.
Fill in drill holes with appropriate filling compound.
Before you move out, have damaged or heavily soiled wallpaper replaced by a professional.
In the bathroom/shower/WC, make sure to keep a sharp eye out for any limescale deposits. Make sure to clean off the grease from any wall tiles in the kitchen.