In case of "sinistre"

Expert
If you have a “sinistre” : fire “incendie”, flood “inondation”. Contact your insurance to make your claim “déclaration”.
Your insurance probably will order a "expertise". It will send an expert to your home to assess damage and he will give its findings to your insurance.
It is possible that your insurance asks you to make a “déclaration circonstanciée”.

“Déclaration circonstanciée” what's that ?
It is a letter in which you simply declare the "circumstances" of the disaster.
Your insurance may also ask you to contact your own contractors for estimates. Send these quotes “devis” to your insurance or expert. Do take no decision until the insurance allows you to do so.

I write a "déclaration circonstanciée" 

In case of theft « vol »
You should go complain to the police or the gendarmerie in your city. They will give you a receipt « récipissé de dépôt », a document that proves that you complained.
You must send the receipt with a letter of declaration to your insurance « en recommandé avec accusé de réception ».
Attention, you have two days to make these steps (but they are « jours ouvrés »). Follow these deadlines for not having (more) problems.

A « jour ouvré » what's that ?
It is a day "worked" and therefore Sundays and holidays do not count.

In all cases claims
Before
Take photos of your valuables, keep all your bills, keep the papers, guarantees anything that proves that you have your items (because once stolen how will your insurance believe you?).

During
If this is a flood « inondation » for example, take pictures to have evidence for your insurance. It may seem a bit extreme but insurers in France have no problem to ask all these "guarantees" and sometimes they can complicate life if you did not.

After
Whether it's a robbery, fire, flood try not to touch or remove objects, places as the expert did not pass.
Gather your evidence : the bills, guarantees, photos.

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