DropBox has recently release his new Sync API SDK both iOS and Android platforms.
Regarding to iOS, although this SDK is addressed to the iOS native platform it’s possible to include the API by using the Xamarin.iOS Binding Project. You can find the DropBox Sync API binding project in the monotouch-bindings repository. It’s included the binding files as well as a sample project.
In order to reference the DropBox Sync API managed library to you Xamarin.iOS project you may create your own binding project, configure it and add the reference or you may execute:
$ cd /<<SomePath>>/monotouch-bindings/DropBoxSync/binding && make all
This command will generate the DropBosSync.iOS.dll library. Try it out by executing the sample project. Just copy the assembly to Visual Studio environment and add it as a reference.