All Android devices offer native synchronization to MS Exchange. On many devices you can choose between Google's Exchange implementation and one from the phone manufacturer.
To sync any Microsoft cloud service or corporate Exchange calendar with Android, you can use Android's built-in Exchange sync. This is provided and updated through the Gmail app (hence the GMail Icon for the account).
Go to Android settings - accounts, there create a new account.
Select Exchange as account type, then make sure to select manual configuration. You can ignore any warning to go back to select "Outlook, Hotmail and Live".
Use m.outlook.com as server for all MS cloud services.
Also, many phone manufacturers (e.g. Samsung, Huawei) offer their own sync (usually called Active Sync or Enterprise) which may offer more features than Google's implementation. The MS Outlook app also offers sync to Android since October 2020, but it's still very buggy.
Limitations of MS Exchange Sync in Android
Some device manufacturers have limited sync of past events, sometimes this is configurable in the account settings or Gmail app settings (coupled with the synchronization of past emails).
Android has not support for attachments in MS Exchange.
Android has not native support for the availability field out-of-office, but we are working on support for proprietary solutions if implemented by the phone manufacturer.
Issues with Samsung's Active Sync
Sync from Android to Outlook not working
Samsung's Active Sync currently fails to sync any new or modified events that have a category stored. The newest version of the Samsung email app (which provides the Active Sync adapter) that still works is 184.108.40.206 available e.g. on APK Mirror.
Exchange Calendar missing after restart on Samsung device in Ultra Power Saving Mode
This is a Samsung firmware bug in all Android versions. Please open the Samsung email app, to re-grant Exchange admin-permissions and re-enable calendar sync in the Exchange account. We reported this bug in verbose detail to Samsung.
Issues with Google's Exchange Sync
Double or Triple entries in MS Exchange calendars
Lately, several users reported duplicate entries, in their Exchange calendars. This can be solved by removing and re-adding the account in Android account settings.
Outlook Sync issues since November 2019 Gmail Update
The November update for the Gmail App (which provides the MS Exchange sync-adapter for Android) seems to break Outlook synchronization. Downgrading to an older Gmail version seems to fix this issue. See here for details answers.microsoft.com
Issues with Outlooks new Sync
Outlook only syncs the values free, busy and tentative, not away/elsewhere.
Outlook does not sync categories.
Outlook does not sync the private status.
All-day events are synced to Outlook one day too long.
MS messes with the description field which breaks aCalendar colors and contact linking.