I need to know the insurance company from which one may take travel insurance during my travel in Europe. I am residing in Amsterdam and am planning to cover Switzerland, Italy and France during my trip. Will the travel insurance cover all the countries or do I need to apply separately. Also if any visitor is coming in from another country, for example India, how do I apply for his health insurance during his stay in Europe????
You could take something like the World Nomads insurance here on TP. Unless you are travelling to the US, most countries are generally included, but it depends on your nationality more than anything else. Have a look at their terms and see if it works for you!
You write you're 'residing' in Amsterdam: find out if you are eligible for regular Dutch insurance. If so, any Dutch company is likely to offer better coverage, fairer terms and -mostly- lower fees than any UK or US company ever would. Independer offers free and neutral comparisons between policies.