Having a traveler’s insurance is something to consider. If you live within the European Union, you need to remember to bring your European Health Insurance Card with you.
Another option would be to get a Camino Card. They offer discounts at a multitude of affiliated accommodations/services on the Camino in addition to a whole set of health care services.
Hopefully you will not encounter any problems, but to offer you some peace of mind: for a lot of the minor problems you may encounter there will be help available for pilgrims (free) by the Spanish health services. In addition you must know that pharmacies can help you out with a lot of other minor problems. See also our page on medications in the health section.
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