In Switzerland there are three layers of insurance that can affect a tenant, and confusing them can leave an important hole at the least expected moment. The first is building insurance, mandatory in most cantons but borne by the landlord, not the tenant. Covers the walls, structure and fixtures of the dwelling. What's inside the floor - your couch, your computer, your clothes - doesn't exist for that insurance.
The second layer is home content insurance (assurance ménage or inventaire du ménage), which covers your personal belongings against fire, theft, water damage or natural disasters. It is not mandatory at the national level, but if your apartment is in flames or someone steals from you and you do not have this insurance, the loss is yours one hundred percent. The third is the private RC (responsabilité civile), which covers the damages you cause to third parties: you break a neighbor's glasses, you spill water due to carelessness and the floor below is damaged, or your child hits an expensive ball. CR does not cover damages that you suffer, only those that you cause.
The usual problem is that insurance companies sell both, ménage and RC, combined in a single contract, which leads one to believe that they are the same thing. They are not. And there is another common misconception: many tenants believe that the building is “already insured” and that this protects them. Building coverage is for the building, not what you keep inside. You can have both coverages with different companies if it suits you financially, although the combined offer is usually reasonably priced. The important thing is to understand what each one covers before you need them.