Liability insurance in Germany is highly recommended. This insurance is called private Haftpflichtversicherung (or Privathaftpflichtversicherung). Do not confuse it with Kfz-Haftpflichtversicherung – liability insurance for drivers. For drivers, this insurance is mandatory and covers the damage they may cause while driving.
Privataftpflichtversicherung is not obligatory by law, but by common sense. Most people in Germany have it, so this topic is for those who have recently arrived.