System notifications have been added in Hiri version 1.3.5. Go to Settings -> Preferences to set whethever you want to receive them for all emails or only for meetings/task reminders.

Linux/snap note
If you're using Hiri snap package, you'll need to have at least version 2.30 of snapd. To check your version:

snap version

At this time version 2.30 is in the candidate channel. To install it:

snap refresh core --candidate

We expect 2.30 to hit the stable channel in the next few days.

+1

Not sure how we can use it if there is no notifications of incoming emails.

Agree with this request.  Calendar reminders are paramount.   Nothing worse than missing a meeting because of no notification :-(  Email notifications are important as well, I get Hiri's thoughts on it and its a great goal.  How about we could set a time for disabling notifications, like when we are working on something important.  I wouldn't want to delay my response to my boss or important client because of a lack of notification.

Is there immediate notification of urgent priority emails (especially when my address in To: - not as concerned if cc'd)

Also calendar/meeting requests inside next 30 minutes & 15-5 minutes meeting reminders to previously accepted meetings (yes, meetings are disruptive but unavoidable if accepted)

Not sure if this is assumed or not, but having an option to enable some sort of audible notification in addition to a visible one would be great.

This. I'd say this is the biggest flaw of Hiri atm and I'd say it should have the highest priority possible. It makes Hiri almost unusable for everyday work. I even got in trouble for this, there was an urgent email in the company and I was working on something else. Since there were no notifications I've continued working and remember to check Hiri only after like 5 hours. Then I realized what I missed.

It could work together with dashboard and show unread emails notification only every 30 minutes or so. But I can't stress it enough that there really HAS to be  some form of notifications

definitly needs native desktop notifications! your concept of non-disturbance is great, but there needs to be some kind of "opt-out" to get "usual" behaviour with hiri

I'm considering purchasing Hiri for work. I'm exploring some email clients (Outlook on WINE, etc) and it needs to have calendar notifications/reminders in it. Preferably the ability to accept calendar invites. This idea is the closest I've found to this request and I didn't want to create a new one. Thanks! (KDE + Gentoo, OS X)

Same for Windows. It's pretty unusable without this. Also there is no way to minimise it to the notifications area.

To stay with your original concept of not checking email too often, I'd like to see notifications limited to only designated contacts. In my case, for example, I need to see emails from my boss immediately. Also, maybe, replies to action requests.

Notification like Slack I suggest with number of unread messages in inbox shown and maybe minutes since first unread e-mail. Or - use color. Green when less than 15 minutes since first unread e-mail, yellow when between 15-30 minutes, and red after 30 minutes.

+1 Ubuntu Gnome

Same for Ubuntu Gnome. Thanks!