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.
Wondershare MobileTrans - Wondershare MobileTrans is a 1-Click Phone to Phone Transfer for you to move data between devices with different operating systems. Your must-have phone Transfer.
Super Backup & Restore - Super Backup & Restore app allows users to backup all their stuff such as pictures, videos, apps, text messages, contacts, call history, and calendar data, etc.
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)
SMS Backup+ - Backup Android SMS, MMS and call log to Gmail / Gcal / IMAP - jberkel/sms-backup-plus
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.