You may use a different computer in all of the places you use computers. If you want to make sure you can work on the same library from any device, you must first make sure to open a free account on the Zotero website
Then open the Zotero desktop application and set your syncing preferences.
1. Open the Tools menu along the top display and select Preferences.
2. If you want Zotero to sync you will have to login to your Zotero account with the user name and password you have created. You can set it to sync automatically, or you can sync manually.

.You can always sync manually at any time by clicking on the green sync button at the far right of the tool bar, as shown below.
