If you are looking to travel to the US as a non-American passport holder with an EU member passport then you must register online at least three days before travel via the Electronic System for Travel Authorisation (ESTA).
Once approved, you are allowed to travel to the US for up to two years without re-registering. Apply for your ESTA here. However, the intention of ESTA is for EU residents living in EU countries to use this system and not for EU residents living in the Cayman Islands.
It is strongly advised that residents of the Cayman Islands obtain a US visa from the US Embassy in Kingston, Jamaica. Appointments can be made here.
US Visitor Visas are not a function of the US Consular Agency and are managed through the US Embassy in Kingston, Jamaica instead. Please note that all UK and Euro ESTA and USA visa holders must now have an electronic readable passport in order to be able to enter the USA.
For more information regarding traveling outside the Cayman Islands, please visit our Important Travel Requirements page where we discuss the countries that are eligible to apply for a U.S. Visa Waiver (also known as ESTA) plus lots more!