Based on your survey responses, we recommend that you set up Office 365 mail services with an Exchange 2013 hybrid deployment. ...

Based on your survey responses, we recommend that you set up Office 365 mail services with an Exchange 2013 hybrid deployment. An Exchange hybrid deployment lets you have some mailboxes in your on-premises Exchange organization and some mailboxes in the Office 365 cloud. You manage all mailboxes — those on-premises and those in the cloud — from your existing on-premises Exchange organization. An Exchange hybrid deployment requires setting up directory synchronization between your on-premises Active Directory and Azure Active Directory, the directory service used by Office 365.