eM Client 4+

Obviously the calendar feature is critical. Right now, accepting a meeting invitation is possible with this emClient but… since there is no calendar feature, the event does not actually get put onto any calendar once accepted. This is a problem - don’t accept invitations from this mail client or you risk losing meetings.

But even more importantly, the app needs to refresh in the background. I should always get a notification if I get an email when I’m not looking at my inbox (as Apple Mail and most other email clients do). And, the app’s badge should show me I have X number of new messages, even before I launch it to read those messages. Why should I have to launch the app in order to see if I actually have any new mail? This puts me into “polling” mode which isn’t productive.

Look forward to upcoming enhancements as the UI is very nice and consistent with the desktop app.

I found the desktop app in 2012. Even back then eM Client was doing things no other mail client was doing, and doing other things better. It's all I've used ever since. There's some early feedback in other reviews about apparent issues with the first release of the iOS app. I'm experiencing some of the same things, but I think it's worth pointing out that they could have turned out an under-baked iOS app any time they wanted. Instead, they waited and gave us something that truly displayed their desire to live up to the desktop experience that made many of us excited to give this app a try. If messages or message notifications are slow to come in right now, you can trust they will prioritize a fix. Just give them sufficient time. In the meantime, go through all the settings. There's a depth of control that is surprising for a mobile mail client, and you just might find something that allows you to fix an issue that appeared to be a bug at first.

We use it as a desktop email client, works very well. The only exception is that it overrides default gmail events privacy, and makes all calendar events public unless the user explicitly marks private checkbox. They refuse to fix this for years.
As per the mobile app, similar weird product management decisions are made. All the major IOS mail apps behave in a certain way, and EM Client does things differently.
Using gmail, while in any folder other than inbox, archive button or swipe will remove the email from said folder. This is true for MS Outlook, Apple Mail and other email apps, but not for EM Client. Archive outside of inbox does NOTHING. You get the info message was archived, but it remains in the folder and nothing has happened.
Another example, all other major email apps will send push notifications and show status badge for all favorite folders, not just inbox. EM Client doesn’t have an option to choose which folders are important to a user to get notifications, even though it has favorite folders.
For us, these are dealbreakers, and mobile app is not used.