Backup utilities may notify you when a backup fails, but don't provide coverage for you forgetting to set the job, or not having permissions to a certain resource. As part of my protection plan, in addition to regular notifications, I like to know if a database HAS NOT been backed up for any reason within the past two days.  The following script accomplishes this.

Note that this utilizes Database Mail and a DBMail profile called "Default DB Mail Profile". These must be manually configured.  More information can be found here.

 --Declarations 
DECLARE @Threshold DATE = CONVERT(DATE, Getdate()-2) --The cutoff from when we consider backups to be current (2 days before the end of today) 
DECLARE @SendEmail BIT = 0 
DECLARE @emailbody VARCHAR(2000) = 'The following databases have not been backed since before ' + CONVERT(VARCHAR(32), @Threshold) + ':<BR/><UL>'
DECLARE @DatabaseName VARCHAR(256)
DECLARE @LastBackupTaken VARCHAR(256) 
DECLARE @UserName VARCHAR(256) 
 
DECLARE emailcursor CURSOR FAST_FORWARD FOR 
  SELECT t1.name                                                              AS databasename, 
         Coalesce(CONVERT(DATE, MAX(t2.backup_finish_date)), 'Not Yet Taken') AS lastbackuptaken, 
         Coalesce(CONVERT(VARCHAR(32), MAX(t2.user_name), 101), 'NA')         AS username 
  FROM   sys.sysdatabases t1 
         LEFT OUTER JOIN msdb.dbo.backupset t2 
           ON t2.database_name = t1.name 
  WHERE  database_name NOT IN ( 'TempDB' ) 
  GROUP  BY t1.name 
  HAVING Coalesce(CONVERT(VARCHAR(32), MAX(t2.backup_finish_date), 101), 'Not Yet Taken') < @Threshold 
  ORDER  BY t1.name 
 
--Open Cursor 
OPEN emailcursor 
 FETCH NEXT FROM emailcursor INTO @DatabaseName, @LastBackupTaken, @UserName 
 WHILE @@FETCH_STATUS = 0 
  BEGIN 
      --- 
      SELECT @SendEmail = 1 
       SELECT @emailbody = @emailbody + '<LI>' + @DatabaseName + ' (' + @LastBackupTaken + ')</LI>' 
       FETCH NEXT FROM emailcursor INTO @DatabaseName, @LastBackupTaken, @UserName 
      ---     
  END 
 CLOSE emailcursor 
 DEALLOCATE emailcursor 
 
SELECT @emailbody = @emailbody + '</UL>' 
 
IF @SendEmail = 1 
  BEGIN 
      EXEC msdb.dbo.Sp_send_dbmail 
        @recipients= 'myemail@address.com', 
        @copy_recipients = '', 
        @subject = 'SERVER Database Backups', 
        @body = @emailbody, 
        @body_format = 'HTML', 
        @profile_name = 'Default DB Mail Profile'; 
  END 
GO