The most important types of insurance in the US are liability insurance and D&O insurance. Liability insurance protects you against claims arising from negligent services or products, while directors and officers insurance protects you as a foreign CxO against direct lawsuits against you personally.
In the US, the use of a broker is strongly recommended, because U.S. insurance markets are complex and heavily regulated at the state level. And similar to taxes, it is helpful if your broker is familiar with both the American and European insurance systems.
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I will write more about this in July/August – check back soon!
By the way: Nothing I write here constitutes legal advice within the meaning of the German Rechtsdienstleistungsgesetzes. The content has been compiled to the best of my knowledge. I am not a lawyer (but I know a few, if necessary).