There are many different longer-stay Visas available, which are considered "National Visas" and are controlled by the sovereign country.
(It's important to note from the git go that meeting all the requirements does not guarantee the issuance of a visa.)
First we'll look at the "Elective Residence" Visa, which we lust after from watching movies such as "Under the Tuscan Sun".
Sounds pretty easy, huh, to say I think I'll just move to Italy? Easier said than done!
First, to get an Elective Residence Visa you must be "retired persons, persons with high self-sustaining incomes and financial assets - who have chosen Italy as the country of permanent residence and who are able to support themselves autonomously, without having to rely on employment while in Italy, whether as dependent employees, as self-employed employees or employees working remotely online. You cannot finance your residence in Italy through any type of work."
The applicant must have a minimum monthly income of Euro 2,596.60 generated from investments.
No work is allowed to be done while in residence in Italy under the Elective Residence Visa. Not even online work.