Q: Can I use Xamarin with NokiaX Since Nokia X (and X+ and XL) is based on Android and Xamarin supports Android, can we use Xamarin to build applications for the new Nokia X phones?
Or is this a different breed of Android.

A: Nokia X (and X+ and XL) is built upon the Android Project V4.1.2 (Level 16). Yes, it's compatible with Xamarin.Android and you just have to select the proper API level.
It's also interesting to know that Nokia uses Xamarin and MVVM Cross for Nokia X's pre-installed apps.

One example of this is the Nokia MixRadio app, which started its life
  as a Windows Phone and Windows Store application and using the power
  of the Xamarin platform, is shipping as a pre-installed app on Nokia X
  devices. To make the most of their existing applications, they
  maximised the use of shared code with Portable Class Libraries and the
  popular third party MVVM library MVVMCross whilst still taking
  advantage of the Nokia X platform's nuances and user interface.

Here the official post from Xamarin.

A: Nokia X service APIs (maps, payment and notification) have also been implemented as free Xamarin components.
https://components.xamarin.com/?search=Nokia
Please also note that Xamarin.Android version 4.12.0 or higher is required!
