Despite all care, damage can occur during a move. The question then is: who pays? An overview of the relevant insurances.
The moving company's insurance
Reputable moving companies have liability insurance covering damage caused during transport through their fault. Ask before the job about the extent of cover and any deductible.
Household and liability insurance
Your household insurance generally covers your belongings, but not always moving damage. Private liability insurance covers damage you cause to third parties. Check your policies before the move.
What is often not covered
Boxes you packed yourself are often excluded from the company's liability – they can't check how you packed. Very valuable items should be insured separately.
In case of damage
Report damage immediately, ideally at handover, and have it recorded in writing. Photograph the damage. The faster you react and the better you document, the simpler the settlement.