While Zotero does not offer a fully-fledged mobile app you can:
You can always find out the latest on Zotero add ons and apps from their mobile support page.
Navigate to https://www.zotero.org/ and login to view your Zotero library. Zotero web pages are built using repsonsive layout so they work on most screen sizes.
Try out the Zotero bookmarklet to save items you encounter while browsing the web on your mobile device to your library. Please note that the bookmarklet is still somewhat experimental and seems to work best for Android devices that iOS devices using Safari.
Zandy is a full-featured interface for Zotero on Android devices. It costs $1.99.
With Zandy, you can browse or search your library, add attachments, add web page citations from your browser, and add or edit notes.
PaperShip for iOS devices, including Mac, allows you to view and edit your Zotero library, import new items into your library, and annotate PDFs. PaperShip also allows easy sharing to social networks.
Note - no longer maintained. 11/2017.
BrowZine provides apps for iOS and Android that allow you to easily locate and browse open access journals and to save citations to Zotero, RefWorks, or EndNote.
The University of Arkansas does not have an institutional license for BrowZine, so our subscription journals are not available through this app. However, it has coverage of sme open access titles, with more content available for the sciences.