Assignment of additional Permissions to Users is most common via a Permission Set. In this guide we’ll demonstrate creating and assigning a permission set using Query All Files, Modify All Data, and Bulk API Hard Delete as an example.
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To do this using the Salesforce Lighting Experience type Permission into the Quick Find box and select Permission Sets

You can either modify an existing permission set or create a new one. Here we'll create a new permission set.

Fill in the Label and API Name fields (e.g. ArchivingPermissions), it's up to you what type of users you want to be able to use this permission set. Save this new permission set.

This will bring you to an overview of the permission set, we want to scroll down and select System Permissions

This will take you to a new page, press the Edit button at the top left:

Scroll down until you see Bulk Api Hard Delete permission, enable it.

Scroll down farther and enable the Modify All Data permission. This will automatically enable several others permissions that Modify All Data relies on.

Go back up the the top and hit Save

This will open a modal asking you to confirm the changes, hit Save again.

Finally we want to enable the Query All Files permission which is an App Permission, use the dropdown at the top left to navigate to App Permissions :

Follow the same process as you did with the other permissions to enable the Query All Files permission.

Now that our permission set has the permissions required for archiving we want to assign it to a user. To do this click Management Assignments at the top left.

This will take you to a page showing all the active assignments of this permission set, there should be none as it is a new permission set. Click the Add Assignment button at the top right hand corner of the page:

This takes you to a page with Users you can assign this permission set to. Select the user you want to have archiving permissions:

Then hit the Next button at the bottom right. You'll be taken to a page asking you to select an expiration date for these user permissions. For now I won't add one:

Click Assign at the bottom right to finish assigning archiving permissions to that user.

You should get a success message like this:

This user now has the Bulk API Hard Delete, Modify All Data, and Query All Files permissions enabled via a permission set. For more information about each of these permissions see the Salesforce Docs below:

If you run into a problem with assigning the permission set to the user and you get the following error:

You will need to define the licence for the permission set when you create it - for example setting the required licence type as Salesforce for a standard licence.

Other similar issues with permission set assignments can happen if you are not assigned the full Permission Set Licence Assignments for a particular managed package to your user - for example your archiving user does not have a CPQ Permission Set Licence Assignment. You will need to fix this by Edit Assignments in the User section.