Best for Exchange users: Mail+ for Outlook
Mail+ for Outlook ($6) follows the Exchange philosophy of putting all of your core needs right into one application: Email, contacts, calendar, and--unique among this app list--Outlook's tasks. It connects to any Microsft Exchange and Outlook account.
It also does a better job than others at handling the formatting found in Outlook messages--like when you get a message from your coworker who uses five different colors in their email to highlight different information. Now you can see it on your iPhone or iPad.
While other apps on this list handle Exchange, I'd put this as the top choice if your core productivity takes place with Microsoft's platform. It's also optimized for the iPhone 6 and 6 Plus.
All in with Google? Go with Gmail
As someone who has also used Android, I can say with certainty the Gmail app for iOS isn't as fast or robust as its Android counterpart. However, if you are a power Gmail user and rely on labels, Google's search prowess, and the Googly design, then go with the Gmail app (free). It works with consumer Gmail and Google Apps for Work.
It supports up to five accounts, so you can check up on your personal and work email. It's updated for the iPhone 6 and 6 Plus so you don't have to contend with an ugly and stretched out interface.
It's still a little slower than what I would like, but it works and functions like Gmail, making it easier to transition from the desktop to mobile when powering through subscription offers, coupons, or (in my case) an excessive amount of PR emails.
If you're a Gmail user who loves the Inbox Zero philosophy, than you should give Gmail's Inbox (free) a shot, too, if you can score an invite.
Best for categorizing clutter: Inky Mail
An email account can quickly fill with a mashup of coupons, travel deals, and shipping notices, so Inky Mail (free) works to filter these out--that way, you're not rummaging through this deluge while looking for that hotel reservation. The app's home screen has 15 different categories for actions and messages, including a unified inbox, personal messages, social, packages, maps, contacts, and subscriptions.
The Filtered Inbox is the most powerful tool for focusing on essential correspondence, as it strips out all the excessive newsletters and weekend sale notices. Once there you can swipe to the right to get back to the app's home screen
The different inboxes have other helpful tools for cleaning things out and offer quick access to your email list's "unsubscribe" links. It also offers a very helpful Today widget, which you can customize to show messages from one of your specific filters or the unified inbox.
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