Azure provides storage solutions for your needs, from single location storage to geo-replicated highly available storage. It also provides different levels of performance. You can also encrypt your information stored in Azure Storage.
You can extend your on-premises storage to the cloud using the StoreSimple service.
If you want to move your storage to the cloud and have a big volume of data, you can use Azure Import/Export or Azure Data Box.
Azure provides solutions for big data and regular data storage, transformation, orchestration, and presentation.
Azure HDInsight provides Big Data analysis compatible with open source solutions.
You can run your relational database workloads in Azure, using Azure SQL Database, Azure SQL Database for MySQL, or Azure SQL Database for PostgreSQL.
You can optimize the costs for your on-premises databases by moving cool and archive data to Azure by using Azure Stretch Database.
Azure Redis Cache provides with an enterprise-level managed cache services for your application.
Azure Table Storage provides simple NoSQL storage based on key-value pairs of data.
Azure Data Lake Store provides optimized storage for HDInsight workloads.
Azure Search is a search as a service for abstracting from your application the details of the implementation of a search engine.
If you need to perform an analysis of data that evolves along the time, Time Series Insight provides you with the tools for focusing on detecting patterns and analyzing the evolution of your data.
Cosmos DB is a multi API NoSQL service that provides geo-replication of your data with guaranteed low latency read and write access.