iMazing - iMazing lets you transfer music, files, messages, apps and more from any iPhone, iPad, or iPod to a computer, Mac or PC. Manage and backup your iOS device simply without iTunes. (was DiskAid)
Titanium Backup - Titanium Backup is a very old backup app that has been around since the early days of Android. In fact, it's one of the few apps that covers all versions from 1. Read more about Titanium Backup.
i-FunBox - General file management software for iPhone and other Apple products
oandbackup - Make backups of selected apps on your device and restore from those backups.
iExplorer - iExplorer lets you easily transfer music, playlists, text messages, files and more from any iPhone, iPod, or iPad to a Mac or Windows computer and iTunes.
SyncDroid - SyncDroid is basically and iOS and Android smartphone manager that lets the users of both smartphones to transfer their mobile content from one smartphone to another within few simple steps.