Transactional Replication from SQL Server to Azure SQL
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Below is the article  written by Jean-Yves (JY) Devant, a Senior Program Manager for SQL Server.

Copying Entire Article ,so to avoid link rot

This new capacity of Transactional Replication is available starting with the following versions\updates of SQL Server:

Community Technology Preview (CTP) 3.0 of SQL Server 2016
SQL Server 2014 Service Pack 1 Cumulative Update 3
SQL Server 2014 RTM Cumulative Update 10
SQL Server 2012 Service Pack 2 Cumulative Update 8


This enables two main scenarios:

Migrate your data to Azure SQL DB with no downtime.
Bridge SQL Server on-premises/on VMs to Azure SQL DB.


The way we enable Azure SQL DB as a subscriber is by extending existing logic of Transactional Replication. From the perspective of your experience with it, the only difference you’ll notice is when you create a subscriber: you give the URL to your Azure SQL DB instead of giving the name of a server or an instance. Because we designed the feature so you have a seamless experience, and you already know about Transactional Replication, the learning curve to use this capacity is 0. It will be very easy and quick for you to start replicating your data to Azure SQL DB.

How does it work? There is "no replication" service per say in Azure SQL DB. The actual replication of data is performed through the Distribution agents.

The chart below shows the architecture from a high level.