With SharePoint 2013, a new feature called Site Mailbox has been delivered by Microsoft.
This feature allows integration between SharePoint 2013 and Exchange 2013. It is available for both On Premises and Online (Office 365) deployments.
Email and documents are traditionally kept in two unique and separate data repositories. Most organizations collaborate using both mediums. The challenge is that both email and documents are accessed using different clients. This usually results in a reduction in user productivity and a degraded user experience.
The site mailbox is a new concept in Microsoft Exchange 2013 that attempts to solve this problem. Site mailboxes improve collaboration and user productivity by allowing access to both Microsoft SharePoint 2013 documents and Exchange email using the same client interface. A site mailbox is functionally comprised of SharePoint 2013 site membership (owners and members), shared storage through an Exchange 2013 mailbox for email messages and a SharePoint 2013 site for documents, and a management interface that addresses provisioning and lifecycle needs.
In this post, we will see how to enable and use this Site Mailbox; meaning no infrastructure configuration will be covered here 
The Site Mailbox has a quotas of 5 Gb
Enable Site Mailbox
The first thing to do is to connect to a SharePoint site to enable Site Mailbox.
Once you have be authenticated with a site collection administrator account on your SharePoint site, go to the Site Settings
Then go to the Site Features management (Manage Site Features option below the Site Actions section)
Locate the Site Mailbox feature (which should be available near the end of the page) and enable it (if not yet enabled)
Once enabled, a new content has been automatically added called Mailbox
This is the Team Site Mailbox; if you go there too quickly you will have a message saying the site mailbox is being provisioned
Another message may also appears (as it may take time to apply access rights) saying the mailbox has been created but not yet available
The site mailbox has been created.
It may take up to 30 minutes for you to gain access to the site mailbox. A message will be sent to everyone in the site’s default owners list and default members list when the site mailbox is ready.
A first new email has been automatically received on the site mailbox to provide information on how to reach and use the site mailbox
Access to the Site Mailbox
With Outlook client
If your mailbox is hosted on Exchange 2013 (or Office 365 as it’s for this post), the Site Mailbox will automatically appears on your Outlook client.
Please note this is not possible to add manually Site Mailbox as additional mailbox.
With web browser from SharePoint
Once the Site Mailbox has been successfully provisioned (creation and access rights applied) you can use by clicking on the link available on the left part
This action will open the site mailbox with Outlook Web Access (the first time you may have to define some OWA settings such as Language and Time Zone, as for any ‘normal’ user when using OWA for the first time)
With web browser from OWA
When you are connected to your mailbox with OWA, you can not view the Site Mailbox directly.
You must open another mailbox by opening the menu below your name and choose Open another mailbox
Then enter the site mailbox name; if the name is not automatically provided, you can search through the directory
Then the Site Mailbox will be opened
