You can download Zotero on the Zotero download page. Be sure to also install a Zotero Connector for your browser.
Open the .dmg you downloaded and drag Zotero to the Applications folder. You can then run Zotero from Spotlight, Launchpad, or the Applications folder and add it to your Dock like any other program.
Run the setup program you downloaded.
Linux users may wish to use a package for their distro rather than using the tarball directly. If you download the tarball, unzip it and run
zotero from that directory to start Zotero.
Alternatively, on Ubuntu, move the directory to a location of your choice, run the bundled
set_launcher_icon script (which will update the included desktop file with the full path to the bundled Zotero icon), and then double-click
Zotero (which may display as
zotero.desktop initially). You can also symlink
~/.local/share/applications/ to add it to your launcher.
Zotero should be updated automatically by default, or you can go to the Help menu and select “Check for Updates…” to check manually. You can also always manually install a new version of Zotero over your existing version without losing any data.
If you're currently running Zotero 4.0 for Firefox and wish to upgrade to Zotero 5.0, you'll need to install Zotero 5.0 from the download page along with the new Zotero Connector for Firefox. See A Unified Zotero Experience for more information.