While there is no official Zotero app at present, Zotero's website lists a number of third party apps that work with Zotero.
Cross-Platform
Zotero Reader (allows you to view your Zotero library, including attachments, in Android, iPad and desktop)
For iOS
BibUp (allows you to add books to your library by scanning their ISBN barcodes)
PaperShip (allows you to view and add to your library, read and annotate attachments, and share references)
ZotPad (allows you to view and edit your library, as well as read and annotate attachments).
For Android
Scanner for Zotero (allows you to add books to your library by scanning their ISBN barcodes)
Zandy (view and edit your Zotero library)
Other
Zotfile (a Zotero plugin that syncs your library with your mobile PDF reader and extracts mobile annotations into Zotero notes)