A tax resident of a country with a RESIDENCE-BASED tax system pays taxes on their WORLDWIDE income. However, if you lose tax residency status there (e.g., by moving out), the country will generally stop taxing you.
Several incentives are available for new residents. Employed and self-employed can claim a 70% to 90% tax reduction on their Italian income depnding on the region of residence. Retirees can claim a flat 7% tax rate on their non-Italian income. And hign net-worth individual can use a €100,000-a-year tax payment that takes care of taxation of their non-Italian income of any size.
Italy is part of the European Economic Area (EEA), allowing its citizens to live and work in any other EEA country, plus Switzerland. (The EEA consists of 27 European Union countries plus Iceland, Norway and Liechtenstein.)