I never imagined replacing Apple’s default mail software as my email app on the iPhone, but it just happened. Microsoft’s free Outlook app for iOS is awesome. It’s super-fast. It presents conversations in a clear and easy to understand way. It does a good job at separating conversations between important items and “other”. It also lets you see your entire mailbox if you prefer to do that. It manages attachments intelligently and allows you to connect to Dropbox. It also works effectively with multiple email accounts.
The calendaring functions are also second to none. You access your schedule without leaving the app and easily allows you to schedule meetings and invite guests. You also can quickly send via email the times when you are free during a day (to attend other meetings).
It’s also easy to customize. I changed the default setting such that a left swipe deletes emails, while a right swipe moves them.
Try the app!