I would like to see my Visual Studio Online (in particular my backlog, PBIs and tasks) on my Android phone.
Do you know if there is an app that include this feature?
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Work Item Studio, developed by Microsoft, does that.
Work Item Studio, a Microsoft Garage project, allows you to view, create and edit work items from your projects on Visual Studio Online.
You can also see a list of all your queries, along with the number of results in each of them. Quickly run your queries to see detailed results.
Here are some screenshots from its Google Play space:
Alternatively, I'd suggest you to try browsing your Visual Studio Online account with the browser on your device. If the website follows good principles of responsive design, you might be able to do all you need directly from there.