To-Do List App with Sync, Repeating To-Dos

2Do

has all the features that you require (and more):

• Due dates ✓
• Notes ✓
• Sync to a cloud or web-based service where changes can be made ✓ (Dropbox, iCloud and other)
• Able to repeat to-dos by due date and last-completed date ✓
• Prioritization ✓
• Groups / Filters / Folders such as Business, Personal, School ✓
• Badge for items due today (tab for today)
• Projects (which incorporate other to-do items) ✓

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Todo by Appigo does everything you request:

• Due dates Yes
• Notes Yes, per task and per project
• Sync to a cloud or web-based service where changes can be made Yes, sync with toodledo, icloud, dropbox, or a few other methods
• Able to repeat to-dos by due date and last-completed date Yes
• Prioritization Yes
• Groups / Filters / Folders such as Business, Personal, School Yes
• Badge for items due today Yes, and overdue items
• Projects (which incorporate other to-do items) Yes - project contain subtasks

There's also an iPad version, and a desktop app, and they all sync to each other using toodledo, dropbox, or icloud.