Exchange Online
Common Exchange Online admin gets and sets: mailbox permissions, send as, and calendar permissions.

Common Admin Gets
Get mailbox permissions
Sometimes you just need to know who has access to a mailbox. The below will return the current permissions on the mailbox.
$Identity = username@domain.com
Get-MailboxPermission -identity $Identity
Common Admin Sets
Adding Permission to a mailbox without automapping
In the below example “mailbox@domain.com” - is the UPN of the mailbox that will receive the new permissions. The “usergettingrights@domain.com” is the UPN of the user who will now be able to access the mailbox. The Automapping set to false - will prevent the calendar of the mailbox from mapping to the end user in question.
Add-MailboxPermission -Identity mailbox@domain.com -User usergettingrights@domain.com -AccessRights FullAccess -AutoMapping $false
Adding send as to a mailbox
When granting the full access control to a mailbox. This does NOT include the ability to send mail as the user, or shared mailbox. This permission is a different delegation, called a trustee right. In the below example “mailbox@domain.com” - is the UPN of the mailbox that will receive the new permissions. The “usergettingrights@domain.com” is the UPN of the user who will now be able to send mail as the “mailbox@domain.com” user.
Add-RecipientPermission -Identity mailbox@domain.com -trustee usergettingrights@domain.com -AccessRights SendAs
Add Calendar Permissions
Calendar permissions are dangerous, but you can do those too. It’s important to keep in mind the “send notification to user” option. You should also keep an eye out for changes like this, as they can indicate someone sneaking access to calendar invites.
In the below example the identity param “youownmycalendar@domain.com:\Calendar” indicates the calendar called “calendar” is what permissions are being given to. The user param is who is getting the permissions.
Add-MailboxFolderPermission -identity youownmycalendar@domain.com:\Calendar -User whoownsyourcalendar@domain.com -SendNotificationToUser $false -AccessRights Editor
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