Microsoft Azure is an extraordinary technology and cloud that allows creation of complete environment and SharePoint can benefit out of the Azure platform to provide Private Cloud SharePoint farms to customer and development and testing labs
Collecting in this article several sources on how to host SharePoint on top of Azure

Microsoft Azure Architectures for SharePoint 2013

Planning for SharePoint 2013 on Azure Infrastructure

This link answer several questions regarding monitoring, backup and recovery

Deploying SharePoint 2013 with SQL Server Always On Availability Group in Azure

a.     Phase 1: Configure Azure

b.    Phase 2: Configure Domain Controllers

c.     Phase 3: Configure SQL Server Infrastructure

d.    Phase 4: Configure SharePoint Servers

e.     Phase 5: Enable Always on Availability Group

f.      Phase 5: Create an Availability Group

g.     Create a Cross-Premises Virtual Network Using Configuration Tables

     Using Microsoft Azure Active Directory for SharePoint 2013 Authentication

Other resources for Building Environments

 SharePoint 2013 Disaster Recovery in Microsoft Azure

     Internet Sites in Microsoft Azure using SharePoint 2013 Server