Office 365 Exchange online tenants have the new sharing experience which will sync shared calendars to mobile devices. At this time, it's required to share from Outlook on the web and the recipient needs to accept it using Outlook on the web.
An Outlook user writes:
I need to be able to view other people's Exchange calendars on my iPad. On my iPad I can see one of my Colleagues calendars but all of my colleagues have shared their calendars out to me and given me full owners permissions, but I can't see their calendars.
The reason you can see the one calendar in your mailbox is because you copied the calendar to your mailbox. This is a copy of the calendar as of the day you copied it and it does not sync with the original calendar.
You cannot see other user's calendar's on an iPad (or iPhone, Windows Phone, Android, or BlackBerry) as these devices only sync your own calendars. You will need to copy the calendar to a folder in your mailbox in order to see it on the device.
See How to View a Shared Calendar on a Smartphone for a for a solution that uses published calendars on smartphones and tablets.
The tools below will be able to sync the shared folders in your profile to a folder in your mailbox. i highly recommend using subfolders for the shared folders, preferably in their own top level folder. For example, create a folder called "Shared Folders" then create a calendar in this folder for each calendar you will be syncing.
CalendarOne is an iPhone / iPad app that allows users to view shared calendars. Now does bi-directional sync and supports nested Calendars. Designed to work with Microsoft Exchange 2010.
CodeTwo FolderSync Addin is an Outlook add-in that enables synchronization of folders. The synchronization of data will always take place between selected folders and the user can choose several pairs of folders that need synchronizing. Freeware.
Priasoft has created the first, and currently only toolset for synchronizing Public Folders between an on-premises deployment of Public Folders and Office365. This same solution is also able to support cross-forest synchronization between one or more on-premises environments, or between multiple on-premises deployments and Office365.
Additional Sync tools can be found at
Synchronizing Exchange Mailboxes and Public Folders