I've seen this mentioned on the Apple forums and looks like its meant to be fixed but I'm still having a problem with it:
I have the default Mail app on the iPhone and iPad setup to my outlook.com e-mail account and its set to Push, however I keep finding after a little while the Push stops working and if I go into my mail app it suddenly catches up.
I've been going into the settings and switching the mail settings from Push to Fetch then back to Push, this will fix the issue for a few days if I'm lucky before it stops working again.
I do have the Outlook app installed also but just using that for work e-mails.
Anyone have any ideas on a proper fix or this just an ongoing iOS / Outlook.com bug?