If the accident or claim is not my fault, do I go through my insurance company or the other person’s insurance company?

Report the crash to your insurance company immediately or within the time specified by your policy along with a copy of any police reports or supporting documents. Then report it to the other insurance company.

Typically each company will have its own way of handling a claim. The majority of the time, if it is not your fault, the other insurance company will process the claim. Just keep on top of them to insure your case is being handled promptly and to your satisfaction.