Currently i have a setup where my clients (web apps, iOS app etc) talks to my backend API .NET web app (Nancy) via REST calls. Nothing special.
I now have a requirement
Is there anything in Azure suited to this?
There is another solution in Azure for the microservice approach, one that I think will get a lot of traction, called Service Fabric.
The concern here is that you have an existing application, and it might be a lot harder to adapt it to work with Service Fabric, but it is certainly worth a look.
Here you can find a Service Fabric example of a web service.
Hope this helps!
Best of luck!