I've written the script below to easily identify databases that have not had backups taken within a certain amount of time. This could be useful if you need to build your own monitoring. However, even if you use a third party monitoring tool,as I do, it is helpful to run this query on the server for which I've received an alert.
Many of the servers I manage have a large number of databases. The alerts I receive for overdue backups are database specific. But, in general, if one database is overdue, chances are something has happened on the server. So this query can give me a quick overview.