Depends on your definition of insurance. The concept goes back to at least the Babylonians in something called "bottomry", which is a method of paying for shipping. A loan is given to pay for the cargo, but if there is a mishap and the cargo/ship is lost, the loan doesn't have to be repaid. If the cargo arrives successfully, the person who gave the loan is repaid with significant interest. We have a record from the reign of the Roman emperor Justinian of 12% as the fixed interest rate for these loans.

This is essentially maritime insurance. If you're running ships and one is lost, you don't lose money on the cargo because you bought it with a loan. The loans are potentially high risk, but with the high interest rates it's worth it. Greeks and Romans used the same bottomry concept. The loans could be provided by groups of people as well, companies were formed for the practice. Here's a short mention of it from Plutarch's Life of Cato the Elder:

"He used to loan money also in the most disreputable of all ways, namely, on ships, and his method was as follows. He required his borrowers to form a large company, and when there were fifty partners and as many ships for his security, he took one share in the company himself, and was represented by Quintio, a freedman of his, who accompanied his clients in all their ventures. In this way his entire security was not imperilled, but only a small part of it, and his profits were large."

So maritime insurance goes back a long long time! Further reading: https://www.jstor.org/stable/1011016?seq=3#metadata_info_tab_contents https://archive.org/details/cu31924030231736/page/n7

Life insurance of a sort also existed in Rome, "burial societies" would pool money to pay for funeral expenses. Here's a surviving text detailing the rules for one: https://www.academia.edu/1821169/Associations_funerals_sociality_and_Roman_law_the_collegium_of_Diana_and_Antinous_in_Lanuvium_CIL_14.2112_reconsidered

As for other types and how they spread, that's beyond my field so hopefully someone else will know more.