I'm about to sign up for a SharePoint Online site.
I have a domain name, but can I map this to a SharePoint Online site?
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In your site's administration center, click Domains under Management. You can add your custom domain name here, in a similar fashion to SharePoint Server's Alternate Access Mappings. After adding your domain, click Manage Site Collection on the administration center page, find the site collection you'd like to map to your domain and click Website Domains in the ribbon. Select your domain name and click OK.
There is a step by step walk through here:
They've been 'working on it' since 3rd Jan 2018. This is to change the sharepoint site name.
But, we should have the ability to add a vanity domain name to sharepoint. It's simple feature, they haven't allowed that yet!