There are various Microsoft tools that you can use to help you migrate your database (updated 1/16/19):

Data Migration Assistant (DMA) – enables you to upgrade to a modern data platform by detecting compatibility issues that can impact database functionality on your new version of SQL Server and Azure SQL Database (Singleton and Managed Instance).  It recommends performance and reliability improvements for your target environment.  It allows you to not only move your schema and data, but also uncontained objects from your source server to your target server.  Download version 4.1

Database Experimentation Assistant (DEA) – is a new A/B testing solution for SQL Server upgrades.  It will assist in evaluating a targeted version of SQL for a given workload.  Customers who are upgrading from previous SQL Server versions (SQL Server 2005 and above) to any new version of the SQL Server will be able to use these analysis metrics provided, such as queries that have compatibility errors, degraded queries, query plans, and other workload comparison data, to help them build higher confidence, making it a successful upgrade experience.  Download version 2.6

In short, customers will be able to access and upgrade their databases using DMA, and validate target database’s performance using DEA.

SQL Server Migration Assistant (SSMA) – version 7.11 for Oracle, MySQL, SAP ASE (formerly SAP Sybase ASE), DB2 and Access lets users convert database schema to Microsoft SQL Server schema, upload the schema, and migrate data to the target SQL Server or Azure SQL Database or Azure SQL Data Warehouse (Oracle only).

Azure Database Migration Service (DMS) – this new database migration service simplifies the migration of existing on-premises SQL Server, Oracle, and MySQL databases to Azure, whether your target database is Azure SQL Database, Azure SQL Database Managed Instance or Microsoft SQL Server in an Azure virtual machine.  The automated workflow with assessment reporting, guides you through the necessary changes prior to performing the migration.  When you are ready, the service will migrate the source database to Azure.  Think of this as similar to the SQL Server Migration Assistant (SSMA) and the Data Migration Assistant (DMA), except this is an Azure PaaS so there is no VMs to create or software to install (DMS actually uses SSMA and DMA behind the scenes but DMS orchestrates everything).  For more info see Azure Database Migration Service (DMS).

Also worth mentioning is in Azure Data Factory, where you can use the Copy Activity to copy data (not schema) of different shapes from various on-premises and cloud data sources to Azure.  See Move data by using Copy Activity.

And for those of you looking to migrate data to Azure Cosmos DB (formally called DocumentDB), we have a data migration tool for that.  It can migration from JSON files, MongoDB, SQL Server, CSV files, Azure Table storage, Blob, Amazon DynamoDB, HBase, Azure Cosmos DB collections.  See Azure Cosmos DB: Data migration tool.

One more: Data Warehouse Migration Utility Preview will migrate schema and data from SQL Server and SQL Database to SQL Data Warehouse (SQL DW).

And a few calculators to help you move from on-prem to the cloud: Total Cost of Ownership (TCO) Calculator and Azure SQL Database DTU Calculator and Azure pricing calculator.

Finally, one of our partners, Attunity, has a product called Attunity Replicate for Microsoft Migrations that is a special offering for Microsoft customers to facilitate the migration from a variety of popular commercial and open-source databases to the Microsoft data platform.

Make sure to checkout the Microsoft Database Migration Guide.

More info:

PASS Summit Announcements: DMA/DEA

Migrate from on-prem SQL server to Azure VM IaaS

Migrate from on-prem SQL server to Azure SQL Database

Getting data into Azure Blob Storage

Azure SQL Database pricing

List of different ways / tools to move data from on-premises SQL to Azure SQL Database